Download c. Graph the function. c. Graph the function.

 c. Graph the function.
4-1 Graphing Quadratic Functions
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis
of symmetry, and the x-coordinate of the
vertex.
b. Make a table of values that includes the
vertex.
c. Use this information to graph the function.
ANSWER: a. y-int = 0; axis of symmetry: x = 0; x-coordinate =
0
b.
1. SOLUTION: a. Compare the function
with the standard form of a quadratic function.
Here, a = 3, b = 0 and c = 0.
The y-intercept is 0.
c. The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
2. SOLUTION: a. Compare the function
with the
standard form of a quadratic function.
c. Graph the function.
Here, a = –6, b = 0 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
eSolutions Manual - Powered by Cognero
The x-coordinate of the vertex is
Page 1
.
The y-intercept is 0.
4-1 Graphing
Quadratic
Functions
The equation
of the axis
of symmetry is
.
c.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
3. SOLUTION: c. Graph the function
.
a. Compare the function
with the standard form of a quadratic function.
Here, a = 1, b = –4 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
.
Therefore, x = 2 is the axis of symmetry.
ANSWER: a. y-int = 0; axis of symmetry: x = 0; x-coordinate =
0
b.
c.
The x-coordinate of the vertex is
.
b. Substitute 0, 1, 2, 3 and 4 for x and make the table.
c. Graph the function.
eSolutions Manual - Powered by Cognero
Page 2
The equation of the axis of symmetry is
4-1 Graphing Quadratic Functions
c. Graph the function.
.
Therefore, x = 1.5 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute 0, –1, –1.5, –2 and –3 for x and make
the table.
ANSWER: a. y-int = 0; axis of symmetry: x = 2; x-coordinate =
2
b.
c. Graph the function.
c.
ANSWER: a. y-int = 4; axis of symmetry: x = –1.5; x-coordinate
= –1.5
b.
4. SOLUTION: a. Compare the function
the standard form of a quadratic function.
Here, a = –1, b = –3 and c = 4.
The y-intercept is 4.
with c.
The equation of the axis of symmetry is
.
eSolutions Manual - Powered by Cognero
Page 3
Therefore, x = 1.5 is the axis of symmetry.
c. Graph the function.
4-1 Graphing Quadratic Functions
c.
5. ANSWER: a. y-int = –3; axis of symmetry: x = 0.75; xcoordinate = 0.75
SOLUTION: a. Compare the function
the standard form of a quadratic function.
Here, a = 4, b = –6 and c = –3.
with b.
The y-intercept is –3.
The equation of the axis of symmetry is
.
c.
Therefore, x = 0.75.
The x-coordinate of the vertex is
.
b. Substitute 0, –1, 0.75, 1.5 and 2.5 for x and make
the table.
6. SOLUTION: c. Graph the function.
a. Compare the function
the standard form of a quadratic function.
with Here, a = 2, b = –8 and c = 5.
The y-intercept is 5.
The equation of the axis of symmetry is
.
eSolutions Manual - Powered by Cognero
Therefore, x = 2 is the axis of symmetry.
Page 4
Here, a = 2, b = –8 and c = 5.
The y-intercept is 5.
4-1 Graphing
Quadratic Functions
c.
The equation of the axis of symmetry is
.
Therefore, x = 2 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute 0, 1, 2, 3 and 4 for x and make the table.
Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
7. c. Graph the function.
SOLUTION: Compare the function
standard form of a quadratic function.
with the Here, a = –1, b = 6 and c = –1.
For this function, a = –1, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
ANSWER: a. y-int = 5; axis of symmetry: x = 2; x-coordinate =
2
b.
c.
.
Substitute 3 for x in the function to find the ycoordinate of the vertex.
Therefore, the maximum value of the function is 8.
The domain is all real numbers.
D = {all real numbers}
.
The range is all real numbers less than or equal to
the maximum value.
eSolutions Manual - Powered by Cognero
Page 5
ANSWER: max = 8; D = {all real numbers},
ANSWER: max = 8; D = {all real numbers},
4-1 Graphing Quadratic Functions
Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
8. SOLUTION: Compare the function
standard form of a quadratic function.
7. with the Here, a = 1, b = 3 and c = –12.
SOLUTION: Compare the function
standard form of a quadratic function.
with the For this function, a = 1, so the graph opens up and
the function has a minimum value.
Here, a = –1, b = 6 and c = –1.
The x-coordinate of the vertex is
For this function, a = –1, so the graph opens down
and the function has a maximum value.
.
Substitute –1.5 for x in the function to find the ycoordinate of the vertex.
The x-coordinate of the vertex is
.
Substitute 3 for x in the function to find the ycoordinate of the vertex.
Therefore, the minimum value of the function is –
14.25.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers greater than or equal
to the minimum value.
Therefore, the maximum value of the function is 8.
The domain is all real numbers.
D = {all real numbers}
.
The range is all real numbers less than or equal to
the maximum value.
ANSWER: min = –14.25; D = {all real numbers},
ANSWER: max = 8; D = {all real numbers},
9. SOLUTION: Compare the function
standard form of a quadratic function.
SOLUTION: Here, a = 3, b = 8 and c = 5.
8. eSolutions
Manualthe
- Powered
by Cognero
Compare
function
standard form of a quadratic function.
with the with the Page 6
or this function, a = 3, so the graph opens up and the
function has a minimum value.
D = {all real numbers}, ANSWER: min = –14.25; D = {all real numbers},
4-1 Graphing Quadratic Functions
9. 10. SOLUTION: SOLUTION: Compare the function
standard form of a quadratic function.
with the Compare the function
the standard form of a quadratic function.
with Here, a = 3, b = 8 and c = 5.
Here, a = –4, b = 10 and c = –6.
or this function, a = 3, so the graph opens up and the
function has a minimum value.
For this function, a = –4, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
The x-coordinate of the vertex is
.
.
Substitute 1.25 for x in the function to find the ycoordinate of the vertex.
for x in the function to find the y-
Substitute
coordinate of the vertex.
Therefore, the minimum value of the function is
The domain is all real numbers.
D = {all real numbers}
The range is all real numbers greater than or equal
to the minimum value.
.
Therefore, the maximum value of the function is
0.25
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
ANSWER: max = 0.25; D = {all real numbers},
ANSWER: D = {all real numbers}, 11. BUSINESS A store rents 1400 videos per week at
$2.25 per video. The owner estimates that they will
rent 100 fewer videos for each $0.25 increase in
price. What price will maximize the income of the
store?
10. eSolutions Manual - Powered by Cognero
SOLUTION: SOLUTION: Let x be the number of increase in price and let f (x)
be the income.
Page 7
of symmetry, and the x-coordinate of the
vertex.
ANSWER: max = 0.25; D = {all real numbers},
b. Make a table of values that includes the
vertex.
4-1 Graphing Quadratic Functions
11. BUSINESS A store rents 1400 videos per week at
$2.25 per video. The owner estimates that they will
rent 100 fewer videos for each $0.25 increase in
price. What price will maximize the income of the
store?
c. Use this information to graph the function.
12. SOLUTION: SOLUTION: Let x be the number of increase in price and let f (x)
be the income.
Here, a = 4, b = 0 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
Here, a = −25, b = 125, and c = 3150.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
The function gets maximum value at 2.5.
That is, 2.5 number of increase in price will maximize
the income.
$2.25 + (2.5)(0.25) ≈ $2.88
So, the price of $2.88 per video will maximize the
income of the store.
c. Graph the function.
ANSWER: $2.88
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis
of symmetry, and the x-coordinate of the
vertex.
b. Make a table of values that includes the
vertex.
c. Use this information to graph the function.
eSolutions
Manual - Powered by Cognero
12. ANSWER: a. y-int = 0; axis of symmetry: x = 0; x-coordinate =
0
Page 8
b.
The x-coordinate of the vertex is
.
ANSWER: a. y-int = 0;Quadratic
4-1 Graphing
Functions
axis of symmetry:
x = 0; x-coordinate =
0
b.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
c.
c. Graph the function.
ANSWER: a. y-int = 0; axis of symmetry: x = 0; x-coordinate =
0
b.
13. SOLUTION: a. Compare the function
with the standard form of a quadratic function.
Here, a = –2, b = 0 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
c.
.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
eSolutions Manual - Powered by Cognero
Page 9
14. 4-1 Graphing Quadratic Functions
ANSWER: a. y-int = –5; axis of symmetry: x = 0; x-coordinate =
0
b.
14. SOLUTION: a. Compare the function
with the standard form of a quadratic function.
Here, a = 1, b = 0 and c = –5.
The y-intercept is –5.
c.
The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
15. SOLUTION: c. Graph the function.
a. Compare the function
with the standard form of a quadratic function.
Here, a = 4, b = 0 and c = –3.
The y-intercept is –3.
The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
ANSWER: a. y-int = –5; axis of symmetry: x = 0; x-coordinate =
0
eSolutions
Manual - Powered by Cognero
b.
Page 10
The x-coordinate of the vertex is
.
b. Substitute
–2, –1, 0,Functions
1 and 2 for x and make the
4-1 Graphing
Quadratic
table.
16. SOLUTION: c. Graph the function.
a. Compare the function
with the standard form of a quadratic function.
Here, a = 1, b = 0 and c = 3.
The y-intercept is 3.
The equation of the axis of symmetry is
.
Therefore, x = 0 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
ANSWER: a. y-int = –3; axis of symmetry: x = 0; x-coordinate =
0
b.
c. Graph the function.
c.
eSolutions Manual - Powered by Cognero
16. ANSWER: a. y-int = 3; axis of symmetry: x = 0; x-coordinate =
0
Page 11
b.
The x-coordinate of the vertex is
.
ANSWER: 4-1 Graphing Quadratic Functions
a. y-int = 3; axis of symmetry: x = 0; x-coordinate =
0
b.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
c. Graph the function
.
c.
ANSWER: a. y-int = 5; axis of symmetry: x = 0; x-coordinate =
0
b.
17. SOLUTION: a. Compare the function
standard form of a quadratic function.
with the Here, a = –3, b = 0 and c = 5.
The y-intercept is 5.
The equation of the axis of symmetry is
.
c.
Therefore, x = 0 is the equation of axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –2, –1, 0, 1 and 2 for x and make the
table.
eSolutions Manual - Powered by Cognero
Page 12
18. ANSWER: a. y-int = 8; axis of symmetry: x = 3; x-coordinate =
3
b.
4-1 Graphing Quadratic Functions
18. SOLUTION: a. Compare the function
standard form of a quadratic function.
with the c.
Here, a = 1, b = –6 and c = 8.
The y-intercept is 8.
The equation of the axis of symmetry is
.
Therefore, x = 3 is the equation of the axis of
symmetry.
The x-coordinate of the vertex is
.
b. Substitute 1, 2, 3, 4 and 5 for x and make the table.
19. SOLUTION: c. Graph the function.
a. Compare the function
standard form of a quadratic function.
Here, a = 1, b = –3 and c = –10.
with the
The y-intercept is –10.
The equation of the axis of symmetry is
.
Therefore, x = 1.5 is the equation of the axis of
symmetry.
The x-coordinate of the vertex is
.
b. Substitute 0, 1, 1.5, 2 and 3 for x and make the
table.
ANSWER: a. y-int = 8; axis of symmetry: x = 3; x-coordinate =
3
b.
eSolutions Manual - Powered by Cognero
Page 13
The x-coordinate of the vertex is
.
b. Substitute
0, 1, 1.5, Functions
2 and 3 for x and make the
4-1 Graphing
Quadratic
table.
20. SOLUTION: c. Graph the function.
a. Compare the function
the standard form of a quadratic function.
Here, a = –1, b = 4 and c = –6.
The y-intercept is –6.
The equation of the axis of symmetry is
with .
Therefore, x = 2 is the equation of the axis of
symmetry.
The x-coordinate of the vertex is
.
b. Substitute 0, 1, 2, 3 and 4 for x and make the table.
ANSWER: a. y-int = –10; axis of symmetry: x = 1.5; xcoordinate =1.5
b.
c.
c. Graph the function.
ANSWER: a. y-int = –6; axis of symmetry: x = 2; x-coordinate =
2
b.
eSolutions Manual - Powered by Cognero
20. Page 14
c.
ANSWER: a. y-int = –6; axis of symmetry: x = 2; x-coordinate =
2
4-1 Graphing
Quadratic Functions
b.
c. Graph the function.
c.
ANSWER: a. y-int = 9; axis of symmetry: x = 0.75; x-coordinate
= 0.75
b.
21. c.
SOLUTION: a. Compare the function
the standard form of a quadratic function.
Here, a = –2, b = 3 and c = 9.
The y-intercept is 9.
The equation of the axis of symmetry is
with .
The equation of the axis of symmetry is x = 0.75.
The x-coordinate of the vertex is
Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
.
b. Substitute –1, 0, 0.75, 1.5 and 2.5 for x and make
the table.
22. SOLUTION: Compare the function
form of a quadratic function.
c. Graph the function.
eSolutions Manual - Powered by Cognero
with the standard Here, a = 5, b = 0 and c = 0.
For this function, a = 5, so the graph opens up and
Page 15
the function has a minimum value.
ANSWER: min = 0; D = {all real numbers},
4-1 Graphing Quadratic Functions
Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
23. SOLUTION: Compare the function
with the standard form of a quadratic function.
22. Here, a = –1, b = 0 and c = –12.
SOLUTION: Compare the function
form of a quadratic function.
with the standard For this function, a = –1, so the graph opens down
and the function has a maximum value.
Here, a = 5, b = 0 and c = 0.
The x-coordinate of the vertex is
.
For this function, a = 5, so the graph opens up and
the function has a minimum value.
The x-coordinate of the vertex is
Substitute 0 for x in the function to find the ycoordinate of the vertex.
.
Substitute 0 for x in the function to find the ycoordinate of the vertex.
Therefore, the maximum value of the function is –
12.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
Therefore, the minimum value of the function is 0.
The domain is all real numbers.
D = {all real numbers}
The range is all real numbers greater than or equal
to the minimum value.
ANSWER: max = –12; D = {all real numbers},
ANSWER: min = 0; D = {all real numbers},
24. SOLUTION: 23. Compare the function
standard form of a quadratic function.
SOLUTION: Compare the function
with the standard form of a quadratic function.
eSolutions
Manual
- Powered
Cognero
Here,
a = –1,
b = 0by
and
c = –12.
For this function, a = –1, so the graph opens down
with the Here, a = 1, b = –6 and c = 9.
For this function, a = 1, so the graph opens up and
the function has a minimum value.
Page 16
ANSWER: max = –12; D = {all real numbers},
ANSWER: min = 0; D = {all real numbers},
4-1 Graphing Quadratic Functions
24. 25. SOLUTION: SOLUTION: Compare the function
standard form of a quadratic function.
with the Compare the function
standard form of a quadratic function.
with the Here, a = 1, b = –6 and c = 9.
Here, a = –1, b = –7 and c = 1.
For this function, a = 1, so the graph opens up and
the function has a minimum value.
For this function, a = –1, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
The x-coordinate of the vertex is
.
.
Substitute 3 for x in the function to find the ycoordinate of the vertex.
Substitute –3.5 for x in the function to find the ycoordinate of the vertex.
Therefore, the minimum value of the function is 0.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers greater than or equal
to the minimum value.
Therefore, the maximum value of the function is
13.25.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to the
maximum value.
ANSWER: min = 0; D = {all real numbers},
ANSWER: max = 13.25; D = {all real numbers},
25. 26. SOLUTION: Compare the function
standard form of a quadratic function.
SOLUTION: with the Here, a = –1, b = –7 and c = 1.
For this function, a = –1, so the graph opens down
and the function has a maximum value.
eSolutions Manual - Powered by Cognero
The x-coordinate of the vertex is
Compare the function
standard form of a quadratic function.
Here, a = –3, b = 8 and c = 2.
with the For this function, a = –3, so the graph opens down
Page 17
and the function has a maximum value.
The x-coordinate of the vertex is
max =
ANSWER: max = 13.25; D = {all real numbers},
D = {all real numbers}, 4-1 Graphing Quadratic Functions
26. 27. SOLUTION: SOLUTION: Compare the function
standard form of a quadratic function.
Here, a = –3, b = 8 and c = 2.
Compare the function
standard form of a quadratic function.
with the with the Here, a = –2, b = –4 and c = 5.
For this function, a = –3, so the graph opens down
and the function has a maximum value.
For this function, a = –2, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
The x-coordinate of the vertex is
.
.
Substitute
Substitute –1 for x in the function to find the ycoordinate of the vertex.
for x in the function to find the y-
coordinate of the vertex.
Therefore, the maximum value of the function is 7.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
Therefore, the maximum value of the function is
.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
ANSWER: max = 7; D = {all real numbers},
28. ANSWER: max =
D = {all real numbers}, SOLUTION: Compare the function
with the standard form of a quadratic function.
Here, a = –5, b = 0 and c = 15.
27. eSolutions Manual - Powered by Cognero
SOLUTION: For this function, a = –5, so the graph opens down
Page 18
and the function has a maximum value.
ANSWER: max = 15; D = {all real numbers},
ANSWER: max = 7; D = {all real numbers},
4-1 Graphing Quadratic Functions
28. 29. SOLUTION: SOLUTION: Compare the function
with the standard form of a quadratic function.
Compare the function
standard form of a quadratic function.
Here, a = –5, b = 0 and c = 15.
Here, a = 1, b = 12 and c = 27.
For this function, a = –5, so the graph opens down
and the function has a maximum value.
For this function, a = 1, so the graph opens up and
the function has a minimum value.
with the The x-coordinate of the vertex is
The x-coordinate of the vertex is
.
.
Substitute –6 for x in the function to find the ycoordinate of the vertex.
Substitute 0 for x in the function to find the ycoordinate of the vertex.
Therefore, the maximum value of the function is 15.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
ANSWER: max = 15; D = {all real numbers},
ANSWER: min = –9; D = {all real numbers},
Therefore, the minimum value of the function is –9.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers greater than or equal to
the minimum value.
30. 29. SOLUTION: SOLUTION: Compare the function
standard form of a quadratic function.
with the For this function, a = 1, so the graph opens up and
the function has a minimum value.
with Here, a = –1, b = 10 and c = 30.
The x-coordinate of the vertex is
Compare the function
the standard form of a quadratic function.
Here, a = 1, b = 12 and c = 27.
eSolutions Manual - Powered by Cognero
For this function, a = –1, so the graph opens down
and the function has a maximum value.
.
The x-coordinate of the vertex is
Page 19
ANSWER: min = –9; D = {all real numbers},
ANSWER: max = 55; D = {all real numbers}, 4-1 Graphing Quadratic Functions
30. 31. SOLUTION: SOLUTION: Compare the function
the standard form of a quadratic function.
with Compare the function
the standard form of a quadratic function.
with Here, a = 2, b = –16 and c = –42.
Here, a = –1, b = 10 and c = 30.
For this function, a = 2, so the graph opens up and
the function has a minimum value.
For this function, a = –1, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
The x-coordinate of the vertex is
.
.
Substitute 4 for x in the function to find the ycoordinate of the vertex.
Substitute 5 for x in the function to find the ycoordinate of the vertex.
Therefore, the minimum value of the function is –74.
Therefore, the maximum value of the function is 55.
The domain is all real numbers.
D = {all real numbers}.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to the
maximum value.
The range is all real numbers greater than or equal to
the minimum value.
.
ANSWER: min = –74; D = {all real numbers},
ANSWER: max = 55; D = {all real numbers}, 32. CCSS MODELING A financial analyst determined
that the cost, in thousands of dollars, of producing
31. 2
bicycle frames is C = 0.000025f – 0.04f + 40, where
f is the number of frames produced.
SOLUTION: Compare the function
the standard form of a quadratic function.
with a. Find the number of frames that minimizes cost.
b. What is the total cost for that number of frames?
Here, a = 2, b = –16 and c = –42.
For this function, a = 2, so the graph opens up and
the function has a minimum value.
eSolutions Manual - Powered by Cognero
SOLUTION: a. The x-coordinate of the vertex is:
Page 20
ANSWER: min = –74; D = {all real numbers},
Here, a = –3, b = –9 and c = 2.
The y-intercept is 2.
4-1 Graphing Quadratic Functions
The equation of the axis of symmetry is 32. CCSS MODELING A financial analyst determined
that the cost, in thousands of dollars, of producing
2
.
bicycle frames is C = 0.000025f – 0.04f + 40, where
f is the number of frames produced.
a. Find the number of frames that minimizes cost.
Equation of the axis of symmetry is x = –1.5.
The x-coordinate of the vertex is
b. What is the total cost for that number of frames?
b. Substitute –3, –2, –1.5, –1 and 0 for x and make
the table.
SOLUTION: a. The x-coordinate of the vertex is:
.
The number of frames that minimize the cost is 800.
b. Substitute 800 for f in the function and simplify.
c. Graph the function.
Therefore, the total cost is $24, 000.
ANSWER: a. 800
b. $24,000
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis
of symmetry, and the x-coordinate of the
vertex.
b. Make a table of values that includes the
vertex.
c. Use this information to graph the function.
ANSWER: a. y-int = 2; axis of symmetry: x = –1.5; x-coordinate
of vertex = –1.5
b.
33. SOLUTION: a. Compare the function
the standard form of a quadratic function.
Here, a = –3, b = –9 and c = 2.
with c.
The y-intercept is 2.
The equation of the axis of symmetry is .
eSolutions Manual - Powered by Cognero
Equation of the axis of symmetry is x = –1.5.
Page 21
c.
c. Graph the function.
SOLUTION: ANSWER: a. y-int = –9; axis of symmetry: x = 1.5; x-coordinate
of vertex = 1.5
b.
4-1 Graphing Quadratic Functions
34. a. Compare the function
the standard form of a quadratic function.
with Here, a = 2, b = –6 and c = –9.
The y-intercept is –9.
The equation of the axis of symmetry is
.
c.
Therefore, x = 1.5 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute 0, 1, 1.5, 2 and 3 for x and make the
table.
35. SOLUTION: c. Graph the function.
a. Compare the function
standard form of a quadratic function.
Here, a = –4, b = 5 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
with the .
eSolutions Manual - Powered by Cognero
Equation of the axis of symmetry is x = .
Page 22
The y-intercept is 0.
The equation of the axis of symmetry is
4-1 Graphing Quadratic Functions
.
c.
Equation of the axis of symmetry is x =
.
The x-coordinate of the vertex is
.
b. Substitute
for x and make the
table.
36. SOLUTION: a. Compare the function
standard form of a quadratic function.
Here, a = 2, b = 11 and c = 0.
The y-intercept is 0.
The equation of the axis of symmetry is
c. Graph the function.
with the .
Equation of the axis of symmetry is x = –2.75.
The x-coordinate of the vertex is
.
b. Substitute –4, –3, –2.75, –2.5 and –1.5 for x and
make the table.
ANSWER: a. y -int = 0; axis of symmetry: x =
; x-coordinate
of vertex =
b.
c. Graph the function.
c.
eSolutions Manual - Powered by Cognero
Page 23
The y-intercept is 4.
Equation of the axis of symmetry is
4-1 Graphing
Quadratic Functions
.
c. Graph the function.
Therefore, x = –6 is the axis of symmetry.
The x-coordinate of the vertex is
.
b. Substitute –10, –8, –6, –4 and –2 for x and make
the table.
ANSWER: a. y-int = 0; axis of symmetry: x = –2.75; xcoordinate of vertex = –2.75
b.
c. Graph the function.
c.
ANSWER: a. y-int = 4; axis of symmetry: x = –6; x-coordinate
of vertex = –6
b.
37. SOLUTION: a. Compare the function
the standard form of a quadratic function.
Here, a = 0.25, b = 3 and c = 4.
The y-intercept is 4.
Equation of the axis of symmetry is
with
c.
.
Therefore, x = –6 is the axis of symmetry.
eSolutions
Manual - Powered by Cognero
The x-coordinate of the vertex is
Page 24
.
c.
c. Graph the function
.
4-1 Graphing Quadratic Functions
ANSWER: 38. a. y -int = 6; axis of symmetry: x =
SOLUTION: a. Compare the function
; x-coordinate
of vertex =
with the standard form of a quadratic function.
Here, a = –0.75, b = 4 and c = 6.
The y-intercept is 6.
The equation of the axis of symmetry is
b.
.
Equation of the axis of symmetry is x =
c.
.
The x-coordinate of the vertex is
.
b. Substitute
for x and make the
table.
39. SOLUTION: a. Compare the function
c. Graph the function
.
with the standard form of a quadratic function.
Here, a =
, b = 4 and c =
.
eSolutions Manual - Powered by Cognero
The y-intercept is
or –2.5.
The equation of the axis of symmetry is
Page 25
a. y -int = –2.5; axis of symmetry: x =
the standard form of a quadratic function.
; x-
coordinate of vertex =
4-1 Graphing
Functions
Here, a = Quadratic
, b = 4 and
c=
.
b.
or –2.5.
The y-intercept is
The equation of the axis of symmetry is
.
c.
Therefore, x =
is the axis of symmetry.
The x-coordinate of the vertex is
b. Substitute
.
for x and make
the table.
40. SOLUTION: c. Graph the function.
a. Compare the function
with the standard form of a quadratic function.
Here, a =
,b =
and c = 9.
The y-intercept is 9.
The equation of the axis of symmetry is
.
Therefore, x = 1.75 is the axis of symmetry.
ANSWER: a. y -int = –2.5; axis of symmetry: x =
coordinate of vertex =
; x-
The x-coordinate of the vertex is
.
b. Substitute 0.5, 1.5, 1.75, 2 and 3 for x and make
the table.
b.
eSolutions Manual - Powered by Cognero
Page 26
c. Graph the function.
Therefore, x = 1.75 is the axis of symmetry.
The x-coordinate of the vertex is
.
4-1 Graphing
Quadratic Functions
b. Substitute 0.5, 1.5, 1.75, 2 and 3 for x and make
the table.
41. FINANCIAL LITERACY A babysitting club sits
for 50 different families. They would like to increase
their current rate of $9.50 per hour. After surveying
the families, the club finds that the number of families
will decrease by about 2 for each $0.50 increase in
the hourly rate.
a. Write a quadratic equation that models this
situation.
c. Graph the function.
b. State the domain and range of this function as it
applies to the situation.
c. What hourly rate will maximize the club’s income?
Is this reasonable?
d. What is the maximum income the club can expect
to make?
SOLUTION: a. Let x be the number of increase.
ANSWER: a. y-int = 9; axis of symmetry: x = 1.75; x-coordinate
of vertex = 1.75
b.
b. The function is defined in the interval [0, 25].
Therefore,
.
The maximum value of the function is 484.
Therefore,
.
c.
c. $11; Because the function has a maximum at x =
3, it is in the domain. Therefore, three $0.50
increases is reasonable.
d. The value of the function at x = 3 is 484.
Therefore, the maximum income the club can expect
to make is $484.
ANSWER: a. I(x) = –x2 + 6x + 475
b.
;
41. FINANCIAL LITERACY A babysitting club sits
for 50 different families. They would like to increase
their current rate of $9.50 per hour. After surveying
the families, the club finds that the number of families
will decrease by about 2 for each $0.50 increase in
the hourly rate.
eSolutions Manual - Powered by Cognero
a. Write a quadratic equation that models this
situation.
c. $11; Because the function has a maximum at x =
3, it is in the domain. Therefore, three $0.50
increases is reasonable.
d. $484
Page 27
42. ACTIVITIES Last year, 300 people attended the
Franklin High School Drama Club’s winter play. The
ANSWER: a. $11.50
b. $2645
c. $11; Because the function has a maximum at x =
3, it is in the domain. Therefore, three $0.50
increases is reasonable.
4-1 Graphing Quadratic Functions
d. $484
42. ACTIVITIES Last year, 300 people attended the
Franklin High School Drama Club’s winter play. The
ticket price was $8. The advisor estimates that 20
fewer people would attend for each $1 increase in
ticket price.
CCSS TOOLS Use a calculator to find the maximum or minimum of each function. Round
to the nearest hundredth if necessary.
43. SOLUTION: a. What ticket price would give the greatest income
for the Drama Club?
Enter
as Y1.
KEYSTROKES: Y= 1 2 2
X – 2 1 + 8
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 3 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER b. If the Drama Club raised its tickets to this price,
how much income should it expect to bring in?
SOLUTION: a. Let x be the number of increase.
So, the maximum value of the function is –1.19.
The function gets maximum value at 3.5.
ANSWER: min = –1.19 Therefore, $11.50 will give the greatest income for
the Drama Club.
b. Substitute 3.5 for x in the function and simplify.
44. SOLUTION: Enter
as Y1.
KEYSTROKES: Y= (–) The Drama Club will get $2645.
2 ANSWER: a. $11.50
b. $2645
Fix the left and right bounds.
2
X – 1 + 1 9 KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER CCSS TOOLS Use a calculator to find the maximum or minimum of each function. Round
to the nearest hundredth if necessary.
43. SOLUTION: Enter
as Y1.
eSolutions Manual - Powered by Cognero
KEYSTROKES: Y= 1 2 1 + 8
2
X – 2 So, the maximum value of the function is 23.
ANSWER: Page 28
So, the maximum value of the function is 23.
So, the maximum value of the function is –1.19.
ANSWER: 4-1 Graphing Quadratic Functions
min = –1.19 ANSWER: max = 23 45. 44. SOLUTION: SOLUTION: Enter
Enter
as Y1.
as Y1
2
KEYSTROKES: Y= (–) 2 KEYSTROKES: Y= (–) 8 . 3 X – 1 2
X + 1 4 + 1 9 – 6 Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER So, the maximum value of the function is –0.01
.
So, the maximum value of the function is 23.
ANSWER: max = –0.01
ANSWER: max = 23 46. 45. SOLUTION: SOLUTION: Enter
Enter
as Y1
KEYSTROKES: Y= (–) 8 . 3 2
X + 1 4 as Y1.
– 6 Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER eSolutions
Manual - Powered by Cognero
So, the maximum value of the function is –0.01
.
KEYSTROKES: Y= 9 . 7 3 2
X – 1
– 9
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 3 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER Page 29
So, the maximum value of the function is –13.36.
So, the maximum value of the function is –13.36.
So, the maximum value of the function is –0.01
.
ANSWER: min = –13.36 ANSWER: 4-1 Graphing
Quadratic Functions
max = –0.01
47. 46. SOLUTION: SOLUTION: Enter
Enter
as Y1.
KEYSTROKES: Y= 9 . 7 3 as Y1.
2
KEYSTROKES: Y= 2 8 X – 1
8 – 9
X
– 1 5 – 1 2
Fix the left and right bounds.
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 3 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER So, the maximum value of the function is –13.36.
So, the maximum value of the function is –4.11.
ANSWER: min = –13.36 ANSWER: max = –4.11 48. 47. SOLUTION: SOLUTION: Enter
Enter
as Y1.
KEYSTROKES: Y= 2 8 8 X
– 1 5 – 1 2
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER eSolutions Manual - Powered by Cognero
So, the maximum value of the function is –4.11.
as Y1.
KEYSTROKES: Y= (–) 1 6 – 1 4 – 1 2 X
2
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER Page 30
So, the maximum value of the function is –11.92.
So, the maximum value of the function is –4.11.
So, the maximum value of the function is –11.92.
ANSWER: max = –11.92 ANSWER: 4-1 Graphing
Quadratic Functions
max = –4.11 Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
48. SOLUTION: Enter
as Y1.
KEYSTROKES: Y= (–) 1 6 – 1 4 – 1 2 X
49. 2
SOLUTION: Compare the function
with the standard form of a quadratic function.
Here, a = –5, b = 4 and c = –8.
For this function, a = –5, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
Fix the left and right bounds.
KEYSTROKES: 2nd [CALC] 4 ◄ ◄ ◄ ENTER ► ► ► ► ► ► ► ENTER ENTER .
Substitute 0.4 for x in the function to find the ycoordinate of the vertex.
So, the maximum value of the function is –11.92.
ANSWER: max = –11.92 Therefore, the maximum value of the function is
–7.2.
Determine whether each function has a
maximum or minimum value, and find that
value. Then state the domain and range of the
function.
The domain is all real numbers.
D = {all real numbers}.
SOLUTION: The range is all real numbers less than or equal to the
maximum value.
49. Compare the function
with the standard form of a quadratic function.
Here, a = –5, b = 4 and c = –8.
For this function, a = –5, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
.
Substitute 0.4 for x in the function to find the ycoordinate of the vertex.
eSolutions Manual - Powered by Cognero
ANSWER: max = –7.2; D = {all real numbers},
50. SOLUTION: Compare the function
with the standard form of a quadratic function.
Page 31
Here, a = –4, b = –3 and c = 2.
For this function, a = –4, so the graph opens down
ANSWER: max = –7.2; D = {all real numbers},
4-1 Graphing Quadratic Functions
50. ANSWER: max = 2.5625; D = {all real numbers},
51. SOLUTION: Compare the function
with the standard form of a quadratic function.
Here, a = –4, b = –3 and c = 2.
For this function, a = –4, so the graph opens down
and the function has a maximum value.
The x-coordinate of the vertex is
SOLUTION: Compare the function
with the standard form of a quadratic function.
Here, a = 6, b = 3 and c = –9.
For this function, a = 6, so the graph opens up and
the function has a minimum value.
.
The x-coordinate of the vertex is
Substitute
.
for x in the function to find the y-
Substitute –0.25 for x in the function to find the ycoordinate of the vertex.
coordinate of the vertex
.
Therefore, the minimum value of the function is –
9.375.
Therefore, the maximum value of the function is
2.5625.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to the
maximum value.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers greater than or equal to
the minimum value.
ANSWER: min = –9.375; D = {all real numbers},
ANSWER: max = 2.5625; D = {all real numbers},
52. 51. SOLUTION: SOLUTION: Compare the function
with the standard form of a quadratic function.
Here, a = 6, b = 3 and c = –9.
For this function, a = 6, so the graph opens up and
eSolutions Manual - Powered by Cognero
the function has a minimum value.
The x-coordinate of the vertex is
Compare the function
standard form of a quadratic function.
with the Here, a = –4, b = 2 and c = –5.
For this function, a = –4, so the graph opens down
and the function has a maximum value.
Page 32
The x-coordinate of the vertex is
.
ANSWER: min = –9.375; D = {all real numbers},
4-1 Graphing Quadratic Functions
ANSWER: max = –4.75; D = {all real numbers},
52. 53. SOLUTION: Compare the function
standard form of a quadratic function.
SOLUTION: with the with the
Compare the function
standard form of a quadratic function.
Here, a = –4, b = 2 and c = –5.
For this function, a = –4, so the graph opens down
and the function has a maximum value.
Here, a =
The x-coordinate of the vertex is
, b = 6 and c = –10.
For this function, a =
.
, so the graph opens up and
the function has a minimum value.
The x-coordinate of the vertex is
Substitute 0.25 for x in the function to find the ycoordinate of the vertex.
.
Substitute –4.5 for x in the function to find the ycoordinate of the vertex.
Therefore, the maximum value of the function is –
4.75.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
Therefore, the minimum value of the function is –
23.5.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers greater than or equal
to the minimum value.
ANSWER: max = –4.75; D = {all real numbers},
ANSWER: min = –23.5; D = {all real numbers},
53. SOLUTION: with the
Compare the function
standard form of a quadratic function.
Here, a =
54. SOLUTION: , b = 6 and c = –10.
Manual - Powered by Cognero
eSolutions
For this function, a =
, so the graph opens up and
Compare the function
standard form of a quadratic function.
with the
Page 33
max =
ANSWER: min = –23.5; D = {all real numbers},
4-1 Graphing Quadratic Functions
; D = {all real numbers},
Determine the function represented by each
graph.
54. SOLUTION: with the
Compare the function
standard form of a quadratic function.
, b = 4 and c = –8.
Here, a =
55. For this function, a =
, so the graph opens down
SOLUTION: Given graph is a parabola. Therefore, the function
2
and the function has a maximum value.
must be in the form of f (x) = ax + bx + c.
Substitute the points (0, –5) and (2, –9) in the
function.
The x-coordinate of the vertex is
.
for x in the function to find the y-
Substitute
coordinate of the vertex.
The vertex of the graph is (2,–9).
Therefore, the maximum value of the function is
Therefore, the x-coordinate of the vertex is
.
.
The domain is all real numbers.
D = {all real numbers}.
The range is all real numbers less than or equal to
the maximum value.
Substitute –4a for b in the first equation and solve for
a.
Substitute 1 for a in the second equation and solve
for b.
ANSWER: max =
; D = {all real numbers},
Therefore, the required function is
2
f(x) = x – 4x – 5.
eSolutions
Manual - Powered
by Cognero
Determine
the function
represented
by each
Page 34
graph.
ANSWER: f(x) = x – 4x – 5
2
for b.
4-1 Graphing Quadratic Functions
Therefore, the required function is
2
f(x) = x – 4x – 5.
Therefore, the required function is
ANSWER: 2
f(x) = x + 2x – 6.
2
f(x) = x – 4x – 5
ANSWER: 2
f(x) = x + 2x – 6
56. SOLUTION: Given graph is a parabola. Therefore, the function
2
must be in the form of f (x) = ax + bx + c.
57. SOLUTION: Given graph is a parabola. Therefore, the function
2
Substitute the points (–1, –7) and (0, –6) in the
function.
must be in the form of f (x) = ax + bx + c.
Substitute the points (0, 8) and (3, –1) in the function.
The vertex of the graph is (–1,–7).
Therefore, the x-coordinate of the vertex is
The vertex of the graph is (3,–1).
.
Therefore, the x-coordinate of the vertex is
.
Substitute 2a for b in the first equation and solve for
a.
Substitute –6a for b in the first equation and solve for
a.
Substitute 1 for a in the second equation and solve
for b.
Substitute 1 for a in the second equation and solve
for b.
eSolutions
Manual -the
Powered
by Cognero
Therefore,
required
function
2
f(x) = x + 2x – 6.
is
Therefore, the required function is
2
f(x) = x – 6x + 8.
Page 35
Therefore, the required function is
2
f(x) = x + 2x – 6.
ANSWER: 4-1 Graphing Quadratic Functions
2
f(x) = x + 2x – 6
Therefore, the required function is
2
f(x) = x – 6x + 8.
ANSWER: 2
f(x) = x – 6x + 8
58. MULTIPLE REPRESENTATIONS Consider f
2
2
(x) = x – 4x + 8 and g(x) = 4x – 4x + 8.
a. TABULAR Make a table of values for f (x) and g
(x) if
b. GRAPHICAL Graph f (x) and g(x).
57. SOLUTION: Given graph is a parabola. Therefore, the function
must be in the form of f (x) = ax + bx + c.
c. VERBAL Explain the difference in the shapes of
the graphs of f (x) and g(x). What value was changed
to cause this difference?
Substitute the points (0, 8) and (3, –1) in the function.
d. ANALYTICAL Predict the appearance of the
graph of h(x) = 0.25x – 4x + 8. Confirm your
prediction by graphing all three functions if
2
2
SOLUTION: a.
The vertex of the graph is (3,–1).
Therefore, the x-coordinate of the vertex is
.
Substitute –6a for b in the first equation and solve for
a.
b.
Substitute 1 for a in the second equation and solve
for b.
Therefore, the required function is
2
f(x) = x – 6x + 8.
ANSWER: c. Sample answer: g(x) is much narrower than f (x).
The value of a changed from 1 to 4.
d. Sample answer: The graph of h(x) will be wider
than f (x).
2
f(x) = x – 6x + 8
eSolutions
Manual - Powered
by Cognero
REPRESENTATIONS
58. MULTIPLE
Consider f
2
2
(x) = x – 4x + 8 and g(x) = 4x – 4x + 8.
Page 36
c. Sample answer: g(x) is much narrower than f (x).
The value Quadratic
of a changed
from 1 to 4.
4-1 Graphing
Functions
d. Sample answer: The graph of h(x) will be wider
than f (x).
59. VENDING MACHINES Omar owns a vending
machine in a bowling alley. He currently sells 600
cans of soda per week at $0.65 per can. He
estimates that he will lose 100 customers for every
$0.05 increase in price and gain 100 customers for
every $0.05 decrease in price. (Hint: The charge
must be a multiple of 5.)
a. Write and graph the related quadratic equation for
a price increase.
ANSWER: a.
b. If Omar lowers the price, what price should he
charge in order to maximize his income?
c. What will be his income per week from the
vending machine?
SOLUTION: a. Let x be the number of increase.
Convert the price into cents.
b.
c. Sample answer: g(x) is much narrower than f (x).
The value of a changed from 1 to 4.
d. Sample answer: The graph of h(x) will be wider
thanf (x).
b. Let x be the number of decrease.
Convert the price into cents.
The function is maximum at 3.5.
eSolutions Manual - Powered by Cognero
59. VENDING MACHINES Omar owns a vending
machine in a bowling alley. He currently sells 600
cans of soda per week at $0.65 per can. He
Therefore, Omar should charge (65 – 5(3.5) = 475)
45 cents or 50 cents.
Page 37
c. Suppose the number of decrease is 3. Then his
income is:
The function is maximum at 3.5.
4-1 Graphing Quadratic Functions
Therefore, Omar should charge (65 – 5(3.5) = 475)
45 cents or 50 cents.
c. Suppose the number of decrease is 3. Then his
income is:
f(x) = (65 – 5(3))(600 + 100(3))
= 50 × 900
= 45000 cents or $450
Suppose the number of decrease is 4. Then his
income is:
b. Omar can charge at 45 cents or 50 cents.
c. $450 per week
60. BASEBALL Lolita throws a baseball into the air
and the height h of the ball in feet at a given time t in
seconds after she releases the ball is given by the
function
2
h(t) = –16t + 30t + 5.
a. State the domain and range for this situation.
b. Find the maximum height the ball will reach.
f(x) = (65 – 5(4))(600 + 100(4))
= 45 × 1000
= 45000 cents or $450
Omar’s income per week from the vending machine
is $450.
ANSWER: a. f (x) = 39,000 – 3500x – 500x
2
SOLUTION: a. Time t is always positive. So, t is greater than or
equal to zero.
The t-intercept of the function is 2.09.
Therefore,
.
The x-coordinate of the vertex is
The maximum of the function.
b. Omar can charge at 45 cents or 50 cents.
c. $450 per week
60. BASEBALL Lolita throws a baseball into the air
and the height h of the ball in feet at a given time t in
seconds after she releases the ball is given by the
function
2
h(t) = –16t + 30t + 5.
a. State the domain and range for this situation.
b. Find the maximum height the ball will reach.
SOLUTION: a. Time t is always positive. So, t is greater than or
equal to zero.
The t-intercept of the function is 2.09.
Therefore,
.
b. The maximum height the ball will reach is 19.0625
ft.
ANSWER: a.
b. 19.0625 ft
61. CCSS CRITIQUE Trent thinks that the function f
(x) graphed below, and the function g(x) described
next to it have the same maximum. Madison thinks
that g(x) has a greater maximum. Is either of them
correct? Explain your reasoning.
Therefore,
.
eSolutions Manual - Powered by Cognero
The x-coordinate of the vertex is
Page 38
ANSWER: a.
4-1 Graphing
Quadratic Functions
b. 19.0625 ft
61. CCSS CRITIQUE Trent thinks that the function f
(x) graphed below, and the function g(x) described
next to it have the same maximum. Madison thinks
that g(x) has a greater maximum. Is either of them
correct? Explain your reasoning.
ANSWER: Sample answer: Always; the coordinates of a
quadratic function are symmetrical, so x-coordinates
equidistant from the vertex will have the same ycoordinate.
63. CHALLENGE The table at the right represents
some points on the graph of a quadratic function.
a. Find the values of a, b, c, and d.
b. What is the x-coordinate of the vertex?
c. Does the function have a maximum or a
minimum?
SOLUTION: Sample answer: Madison. Sample answer: f (x) has a
maximum of –2. g(x) has a maximum of 1. When
Trent found the x-coordinate of the vertex, he
multiplied two negatives and mistakenly kept a
negative.
ANSWER: Sample answer: Madison. When Trent found the xcoordinate of the vertex, he multiplied two negatives
and mistakenly kept a negative.
62. REASONING Determine whether the following is
sometimes, always, or never true. Explain your
reasoning.
In a quadratic function, if two x-coordinates are
equidistant from the axis of symmetry, then they
will have the same y-coordinate.
SOLUTION: Sample answer: Always; the coordinates of a
quadratic function are symmetrical, so x-coordinates
equidistant from the vertex will have the same ycoordinate.
ANSWER: Sample answer: Always; the coordinates of a
quadratic function are symmetrical, so x-coordinates
equidistant from the vertex will have the same ycoordinate.
SOLUTION: a. Substitute the points (–20, –377), (–5, –2), and (–1,
22) from the table in the general quadratic function f
2
(x) = ax + bx + c to get a system of three equations
in three variables.
400a – 20b + c = –377
25a – 5b + c = –2
a – b + c = 22
The solution of the system is a = –1, b = 0, and c =
2
23. So, the quadratic function is f (x) = –x + 23.
Substitute (5, a – 24) into f (x) to find a.
Substitute (7, –b) into f (x) to find b.
63. CHALLENGE The table at the right represents
quadratic function.
eSolutions
- Powered
by Cognero
someManual
points
on the graph
of a
Page 39
a. Find the values of a, b, c, and d.
Substitute (c, –13) into f (x) to find c.
4-1 Graphing
Quadratic Functions
Substitute (7, –b) into f (x) to find b.
ANSWER: a. a = 22; b = 26; c = –6; d = 2
b. 0
c. maximum
64. OPEN ENDED Give an example of a quadratic
function with a
a. maximum of 8.
Substitute (c, –13) into f (x) to find c.
b. minimum of –4.
c. vertex of (–2, 6).
SOLUTION: a. Sample answer: f (x) = –x2 + 8
Substitute (d – 1, a) or (d – 1, 22) into f (x) to find d.
b. Sample answer: f (x) = x – 4
2
c. Sample answer: f (x) = x2 + 4x + 10
ANSWER: 2
a. Sample answer: f (x) = –x + 8
b. Sample answer: f (x) = x2 – 4
2
c. Sample answer: f (x) = x + 4x + 10
So, a = 22, b = 26, c = –6, and d = 2.
b. Because b = 0, the x-coordinate of the vertex is 0.
c. For this function, a = –1, so the graph opens down
and the function has a maximum value.
ANSWER: a. a = 22; b = 26; c = –6; d = 2
b. 0
c. maximum
64. OPEN ENDED Give an example of a quadratic
function with a
a. maximum of 8.
65. WRITING IN MATH Why can the discriminant be
used to confirm the number and the type of solutions
of a quadratic equation?
SOLUTION: Sample answer: If the discriminant is positive, the
Quadratic Formula will result in two real solutions
because you are adding and subtracting the square
root of a positive number in the numerator of the
expression. If the discriminant is zero, there will be
one real solution because you are adding and
subtracting the square root of zero. If the
discriminant is negative, there will be two complex
solutions because you are adding and subtracting
the square root of a negative number in the
numerator of the expression.
b. minimum of –4.
ANSWER: Sample answer: If the discriminant is positive, the
Quadratic Formula will result in two real solutions
because you are adding and subtracting the square
root of a positive number in the numerator of the
expression. If the discriminant is zero, there will be
one real solution because you are adding and
Page 40
subtracting the square root of zero. If the
discriminant is negative, there will be two complex
solutions because you are adding and subtracting
c. vertex of (–2, 6).
SOLUTION: a. Sample answer: f (x) = –x2 + 8
Manual - Powered by Cognero
eSolutions
2
b. Sample answer: f (x) = x – 4
ANSWER: 2
a. Sample answer: f (x) = –x + 8
2
b. Sample Quadratic
answer: f (x)Functions
=x –4
4-1 Graphing
2
c. Sample answer: f (x) = x + 4x + 10
65. WRITING IN MATH Why can the discriminant be
used to confirm the number and the type of solutions
subtracting the square root of zero. If the
discriminant is negative, there will be two complex
solutions because you are adding and subtracting
the square root of a negative number in the
numerator of the expression.
66. Which expression is equivalent to
of a quadratic equation?
SOLUTION: A
Sample answer: If the discriminant is positive, the
Quadratic Formula will result in two real solutions
because you are adding and subtracting the square
root of a positive number in the numerator of the
expression. If the discriminant is zero, there will be
one real solution because you are adding and
subtracting the square root of zero. If the
discriminant is negative, there will be two complex
solutions because you are adding and subtracting
the square root of a negative number in the
numerator of the expression.
B
C
D
SOLUTION: ANSWER: Therefore, option B is the correct answer.
Sample answer: If the discriminant is positive, the
Quadratic Formula will result in two real solutions
because you are adding and subtracting the square
root of a positive number in the numerator of the
expression. If the discriminant is zero, there will be
one real solution because you are adding and
subtracting the square root of zero. If the
discriminant is negative, there will be two complex
solutions because you are adding and subtracting
the square root of a negative number in the
numerator of the expression.
ANSWER: B
67. SAT/ACT The price of coffee beans is d dollars for
6 ounces, and each ounce makes c cups of coffee. In
terms of c and d, what is the cost of the coffee
beans required to make 1 cup of coffee?
F
G
66. Which expression is equivalent to
H A
J 6cd
B
K C
SOLUTION: D
The cost of the 1 ounce coffee beans is
.
SOLUTION: ounce of coffee beans is need to make 1cup of
coffee.
Therefore, the cost of the coffee bean required to
Therefore,
the correct
eSolutions
Manual -option
PoweredBbyisCognero
ANSWER: B
answer.
make 1 cup of coffee is
.
Page 41
make 1 cup of coffee is
Therefore, option B is the correct answer.
ANSWER: 4-1 Graphing
Quadratic Functions
B
.
ANSWER: K
67. SAT/ACT The price of coffee beans is d dollars for
6 ounces, and each ounce makes c cups of coffee. In
terms of c and d, what is the cost of the coffee
beans required to make 1 cup of coffee?
68. SHORT RESPONSE Each side of the square base
of a pyramid is 20 feet, and the pyramid’s height is
90 feet. What is the volume of the pyramid?
SOLUTION: Volume of a right regular pyramid is
F
.
G
Base area = 20 × 20 = 400 ft
2
H J 6cd
3
Therefore, the volume of the pyramid is 12000 ft .
K ANSWER: SOLUTION: 12,000 ft
The cost of the 1 ounce coffee beans is
69. Which ordered pair is the solution of the following
system of equations?
.
ounce of coffee beans is need to make 1cup of
coffee.
Therefore, the cost of the coffee bean required to
make 1 cup of coffee is
3
3x – 5y = 11
3x – 8y = 5
A (2, 1)
.
B (7, –2)
ANSWER: K
C (7, 2)
68. SHORT RESPONSE Each side of the square base
of a pyramid is 20 feet, and the pyramid’s height is
90 feet. What is the volume of the pyramid?
D
SOLUTION: Volume of a right regular pyramid is
SOLUTION: Subtract the second equation from the first equation.
.
Base area = 20 × 20 = 400 ft
2
Substitute 2 for y in the first equation and solve for x.
3
Therefore, the volume of the pyramid is 12000 ft .
eSolutions Manual - Powered by Cognero
ANSWER: 3
The solution is (7, 2).
Therefore, option C is the correct answer.
ANSWER: Page 42
Therefore, the volume of the pyramid is 12000 ft .
The solution is (7, 2).
Therefore, option C is the correct answer.
ANSWER: 4-1 Graphing Quadratic Functions
3
12,000 ft
ANSWER: C
69. Which ordered pair is the solution of the following
system of equations?
Find the inverse of each matrix, if it exists.
70. 3x – 5y = 11
3x – 8y = 5
SOLUTION: A (2, 1)
Let
B (7, –2)
.
det (A) = 5
C (7, 2)
D
SOLUTION: Subtract the second equation from the first equation.
ANSWER: Substitute 2 for y in the first equation and solve for x.
The solution is (7, 2).
Therefore, option C is the correct answer.
71. ANSWER: C
SOLUTION: Let
Find the inverse of each matrix, if it exists.
.
det (A) = –24
70. SOLUTION: Let
.
det (A) = 5
ANSWER: eSolutions Manual - Powered by Cognero
72. Page 43
SOLUTION: ANSWER: 45
4-1 Graphing Quadratic Functions
74. 72. SOLUTION: SOLUTION: Let
.
det (A) = 14
ANSWER: 22
75. SOLUTION: ANSWER: Evaluate each determinant.
73. SOLUTION: ANSWER: 0
76. MANUFACTURING The Community Service
Committee is making canvas tote bags and leather
tote bags for a fundraiser. They will line both types
of bags with canvas and use leather handles on both.
For the canvas bags, they need 4 yards of canvas
and 1 yard of leather. For the leather bags, they need
3 yards of leather and 2 yards of canvas. The
committee leader purchased 56 yards of leather and
104 yards of canvas.
ANSWER: 45
a. Let c represent the number of canvas bags, and let
represent the number of leather bags. Write a system of inequalities for the number of bags that
can be made.
74. b. Draw the graph showing the feasible region.
SOLUTION: c. List the coordinates of the vertices of the feasible
region.
d. If the club plans to sell the canvas bags at a profit
of $20 each and the leather bags at a profit of $35
each, write a function for the total profit on the bags.
eSolutions
Manual - Powered by Cognero
ANSWER: Page 44
e . How can the club make the maximum profit?
c. List the coordinates of the vertices of the feasible
region.
4-1 Graphing
Quadratic Functions
d. If the club plans to sell the canvas bags at a profit
of $20 each and the leather bags at a profit of $35
each, write a function for the total profit on the bags.
e . How can the club make the maximum profit?
f. What is the maximum profit?
The club makes the maximum profit if they produce
20 canvas tote bags and 12 leather tote bags.
f. The maximum profit is $820.
ANSWER: a.
b.
SOLUTION: a.
b.
c. (0, 0), (26, 0), (20, 12),
d.
e. Make 20 canvas tote bags and 12 leather tote
bags.
f. $820
c. The vertices of the solution region is (0, 0), (26, 0),
(20, 12) and
.
State whether each function is a linear function.
Write yes or no. Explain.
2
77. y = 4x – 3x
d. The optimal function is
.
e . Substitute the points (0, 0), (26, 0), (20, 12) and
in the function.
SOLUTION: No. It cannot be written as y = mx + b.
ANSWER: No; it cannot be written as y = mx + b.
78. y = –2x – 4
SOLUTION: Yes. It is written in y = mx + b form.
The club makes the maximum profit if they produce
20 canvas tote bags and 12 leather tote bags.
f. The maximum profit is $820.
Manual - Powered by Cognero
eSolutions
ANSWER: Yes; it is written in y = mx + b form.
79. y = 4
SOLUTION: Yes. It is written in y = mx + b form, m = 0.
ANSWER: ANSWER: Page 45
Yes. It is written in y = mx + b form.
ANSWER: 4-1 Graphing
Quadratic Functions
Yes; it is written in y = mx + b form.
ANSWER: –13
2
79. y = 4
82. f (x) = 6x + 18, x = –5
SOLUTION: Yes. It is written in y = mx + b form, m = 0.
SOLUTION: Substitute –5 for x in the function and evaluate.
ANSWER: Yes; it is written in y = mx + b form, m = 0.
Evaluate each function for the given value.
2
80. f (x) = 3x – 4x + 6, x = –2
SOLUTION: Substitute –2 for x in the function and evaluate.
ANSWER: 168
ANSWER: 26
2
81. f (x) = –2x + 6x – 5, x = 4
SOLUTION: Substitute 4 for x in the function and evaluate.
ANSWER: –13
2
82. f (x) = 6x + 18, x = –5
SOLUTION: Substitute –5 for x in the function and evaluate.
eSolutions
Manual - Powered by Cognero
ANSWER: 168
Page 46