Download Answers ( Algebra )

GHSGT Algebra Practice test
Name: _________________________
Choose the correct choice:
1. Each statement describes a transformation of the graph of y = x. Which
statement correctly describes the graph of y = x + 2?
A. It is the graph of y = x translated 2 units to the left.
B. It is the graph of y = x translated 2 units to the right.
C. It is the graph of y = x translated 2 units down.
D. It is the graph of y = x where the slope has been
increased by 1.
2.
A.
B.
C.
D
.
3.
f(x) = x3
g(x) = ?
Which of the following is equal to g(x)?
A. (x + 1)3
B. x3 - 1
C. (x - 1)3
D. x3 + 1
4.
f(x) = |x|
g(x) = ?
The function f(x) has been transformed to give g(x). Which of the following
functions represents g(x)?
A. |x| + 3
B. |x + 3|
C. |x| - 3
D. |x - 3|
5. In which direction must the graph of f(x) =
of g(x) =
A. left
B. right
(x) - 3?
x be shifted to produce the graph
C. down
D. up
6.
The graph of the function f(x) = 1/x is shifted one unit up and one unit to the
right. Which of the following corresponds to the shifted graph?
A.
g(x) =
B.
g(x) =
1
(x + 1)
1
(x - 1)
C.
g(x) =
D.
g(x) =
1
(x + 1)
-1
(x - 1)
-1
+1
+1
-1
7. What is the domain of the function below?
f(x) =
A. 7 < x < 14
B. x < 7, or 7 < x < 14, or x > 14
C. x < 7 or x > 14
1
x2 - 21x + 98
D. x < 7, or 7 < x < 14, or x > 14
8. Express the terms of the following geometric sequence recursively.
A.
B.
C.
D.
9. Describe the rates of change of the following functions.
f(x) = 3x
g(x) = x3
h(x) = x6
A. f(x) and g(x) have a constant rate of change; h(x) has a variable rate of change
B. f(x) has a constant rate of change; g(x) and h(x) have variable rates of change
C. f(x), g(x), and h(x) all have variable rates of change
D. f(x), g(x), and h(x) all have constant rates of change
10. Determine the symmetry of the function graphed below.
A. symmetric about the origin
B. symmetric about the y-axis
C. not symmetric
D. symmetric about the x-axis
11. Which of the following systems of linear equations is equivalent to the matrix
equation below?
A.
B.
C.
D.
12. Add the following.
A.
B.
C.
D.
13. Which is equivalent to the following expression?
-4(a + 9b) - 6b + (4a - 5c) + 13(-4b - 5c)
A. -49b + 8c
B. -94b - 70c
C. -94b - 5c
D. -49b + 60c
14. Multiply the following algebraic expression.
x2 + 2x + 1
x-1
A.
x-1
x+1
B.
x-2
x-1
C. (x + 1)(x - 2)
x-1
D.
x+1
x-1
•
x-2
x+1
15. Which expression is the factored equivalent of x2 - 7x + 12?
A. (x - 3)(x - 4)
B. (x - 1)(x - 3)
C. (x + 3)(x - 4)
D. (x + 4)(x + 3)
16. Which graph best represents the quadratic equation whose solutions are x = -1
and x = -2?
A. W
B. X
C. Y
D. Z
17. Solve for x.
W.
X.
Y.
Z.
A. 24
B. 36
C. no solution
D. 144
18. Solve the following rational equation.
A.
= -10 and
= 20
B.
= 10 and
= -20
C.
= -12 and
= 23
D.
= 12 and
= -23
19. Find the quotient.
A.
B.
C.
D.
20.
Which of the following absolute value functions fits the graph shown above?
A. f(x) = -|3x + 3| + 3
B. f(x) = |3x + 3| - 3
C. f(x) = |3x - 3| + 3
D. f(x) = -|3x - 3| + 3
21. What is the domain of the following graph?
A. {x | -5 < x < 4}
B. {x | -5 < x < 4}
C. {y | -2 < y < 5}
D. {y | -2 < y < 5}
22. Which of the following graphs shows the solution set for the inequality below?
|x + 3| + 9 > 11
A.
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
B.
C.
D.
23. Simplify:
A.
B.
C.
D.
24. What is the x-intercept of the following function?
A. The function does not have an x-intercept.
B. 7
C. 0
D. 1
25. Solve for x in the inequality below.
1
125x + 30
< 25x
A. x < -18
B. x > -18
C. x > 18
D. x < 18
26. Identify the maximum coordinate of the function that is shown and graphed
below.
y = -x2 + 2x - 3
A. (1,2)
B. (-1,2)
C. (1,-2)
D. There is no maximum.
27. The table below shows how many people have purchased coffee at a local coffee
shop each day since its grand opening.
Days Since Grand Opening
Number of People
2
5
7
10
14
10 34 60 114 214
Which equation represents p, the number of people who purchased coffee, after
the shop had been open d days?
A. p = d2 + 6
B. p = d2 + d + 4
C. p = d2 + 0.5d + 5
D. p = d2 - d + 8
28.
A.
B.
C.
D.
29. What is the range of the following function?
y = log(x - 4) + 2
A. (4,
)
B. (-
,-4)
C. (-
,2)
D. (-
,
)
30. Which of the following graphs represents f(x) = x2?
W.
X.
Y.
Z.
A. W
B. X
C. Y
D. Z
31. Each statement describes a transformation of the graph of y = x. Which
statement correctly describes the graph of y = x - 15?
A. It is the graph of y = x where the slope is decreased by 15.
B. It is the graph of y = x translated 15 units up.
C. It is the graph of y = x translated 15 units down.
D. It is the graph of y = x translated 15 units to the left.
32. Each statement describes a transformation of the graph of y = x2. Which
statement correctly describes the graph of y = 4(x - 7)2 - 8?
A. It is the graph of y = x2 vertically compressed, and then translated 8 units up and 7
units to the right.
B. It is the graph of y = x2 vertically stretched, and then translated 7 units up and 8
units to the left.
C. It is the graph of y = x2 vertically stretched, and then translated 8 units down and 7
units to the right.
D. It is the graph of y = x2 vertically compressed, and then translated 7 units down and
8 units to the left.
33. In which direction must the graph of f(x) = x3 be shifted to produce the graph of
g(x) = (x + 9)3?
A. left
B. up
C. right
D. down
34.
f(x) = |x|
g(x) = ?
The function f(x) has been transformed to give g(x). Which of the following
functions represents g(x)?
A. (1/3)|x|
B. 2|x|
C. (1/2)|x|
D. 3|x|
35. If the graph of f(x) = x is shifted 3 units to the left, then what would be the
equation of the new graph?
A. g(x) =
(x - 3)
B. g(x) =
(x) + 3
C. g(x) =
(x) - 3
D. g(x) =
(x + 3)
36. In which direction must the graph of
1
f(x) =
x
be shifted to produce the graph of
g(x) =
1
x-3
?
A. up
B. right
C. down
D. left
37. What is the range of the function below?
f(x) =
A. (-
,
B. [0,
C. [-3,
(x + 3)
)
)
)
D. [0, 3]
38. Express the terms of the following sequence by giving an explicit formula.
9 , 12 , 15 , 18 , 21 , . . .
A. u(n) = 6 + 3n, where n = 1, 2, 3, 4, . . .
B. u(n) = 3 + n, where n = 1, 2, 3, 4 . . .
C. u(n) = 3n, where n = 1, 2, 3, 4, . . .
D. u(n) = n + 3, where n = 1, 2, 3, 4, . . .
39. The Johnsons and the Thompsons both took ski trips. The Johnsons are a family
of seven and the Thompsons are a family of four. They both paid airfare of $300
per person. The Johnsons spent a total of $5,100 for airfare and lodging for the
whole family and the Thompsons spent $1,800 for their airfare and lodging.
If lodging was based solely on the number of nights, not the number of guests,
and if both families spent 6 nights, which of the following statements is correct?
A. The Thompsons spent $400 more per night.
B. The Thompsons $500 more per night.
C. The Johnsons spent $500 more per night.
D. The Johnsons spent $400 more per night.
40. Determine the symmetry of the following function.
A. symmetric about the origin
B. symmetric about the y-axis
C. neither symmetric about the y-axis nor the origin
D. both symmetric about the y-axis and the origin
41.
Which of the following matrix equations is equivalent to the system of equations
above?
A.
B.
C.
D.
42. Multiply the following.
A.
B.
C.
D.
43. Simplify the following expression.
(2x + 4)(x - 4)
A. 2x2 - 12x - 16
B. 2x2 - 4x - 16
C. 2x2 - 4x - 8
D. 2x2 + 4x - 16
44. Add the following algebraic expression.
A.
B.
C.
D.
45. Which expression is the factored equivalent of x2 - 8x + 15?
A. (x - 15)(x - 1)
B. (x - 5)(x + 3)
C. (x + 5)(x + 3)
D. (x - 5)(x - 3)
46. The graph of y = -16t2 + 92t + 3 is given below. Estimate the solutions of 0 = 16t2 + 92t + 3.
A. x = -5.8, 5.8
B. x = 0, 92
C. x = -5.8, 0
D. x = 0, 5.8
47. Solve for x.
A. 121
B. 49
C. no solution
D. 14
48. Solve the following rational equation.
A.
= -66
B.
= 66
C.
= 33
D.
= -33
49. What is the complex conjugate of the following complex number?
-6 - 9i
A. 6 + 9i
B. 6 - 9i
C. -6 + 9i
D. -9 - 6i
50. Which of the following tables corresponds to the graph below?
A.
B.
C.
D.
x
-3
-2
-1
0
1
y
1
2
3
2
1
x
-3
-2
-1
0
1
y
1
2
3
-2
-1
x
-3
-2
-1
0
1
y
1
2
3
4
5
x
-3
-2
-1
0
1
y
-1
-2
-3
-2
-1
Answers ( Algebra )
1. A
2. C
3. D
4. B
5. C
6. B
g(x) =
1
(x - 1)
+1
7. B
8. C
9. C
10. A
11. D
12. D
13. B
14. C
(x + 1)(x - 2)
x-1
15. A
16. A
17. D
18. B
19. D
20. B
21. B
22. C
-9
23. A
24. A
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
25. B
26. A
27. B
28. D
29. D
30. D
31. C
32. C
33. A
34. A
35. D
36. B
37. B
38. A
39. D
40. A
41. B
42. C
43. B
44. B
45. D
46. D
47. B
48. C
49. C
50. A
x
-3
-2
-1
0
1
y
1
2
3
2
1
Explanations ( Algebra )
1. Consider the function y = f(x) with the real number r.
y = f(x) + r
shifts the graph of f(x) up r units
y = f(x) - r
shifts the graph of f(x) down r units
y = f(x + r)
shifts the graph of f(x) left r units
y = f(x - r)
shifts the graph of f(x) right r units
The given equation y = x + 2 can be seen as a function shifted up 2 units or to the left 2
units as shown below.
y = f(x) + 2 = x + 2
or
y = f(x + 2) = (x + 2) = x + 2
Therefore, the only given choice that is correct is the following: it is the graph of y = x
translated 2 units to the left.
2.
3. Consider the graph of a function r(x) with real numbers k and h.
r(x) + k
shifts the graph up k units
r(x) - k
shifts the graph down k units
r(x + h)
shifts the graph to the left h units
r(x - h)
shifts the graph to the right h units
The graph of g(x) is the graph of f(x) shifted up 1 unit.
Therefore, g(x) = f(x) + 1 = x3 + 1.
4. Consider the graph of a function r(x) with real numbers k and h.
r(x) + k
shifts the graph up k units
r(x) - k
shifts the graph down k units
r(x + h)
shifts the graph to the left h units
r(x - h)
shifts the graph to the right h units
The graph of g(x) is the graph of f(x) moved to the left 3 units.
Therefore, g(x) = f(x + 3) = |x + 3|.
5. Notice that g(x) = f(x) - 3. Since the change happens outside the function, we know it
must be a vertical shift. Since the number is being subtracted from x, the graph of g(x)
was created by shifting the graph of f(x) 3 units down.
6. Consider the graph of a function r(x) with real numbers k and h.
r(x) + k
shifts the graph up k units
r(x) - k
shifts the graph down k units
r(x + h)
shifts the graph to the left h units
r(x - h)
shifts the graph to the right h units
The graph of g(x) is the graph of f(x) =
unit to the right. Therefore, g(x)=
1
x
1
(x - 1)
shifted 1 unit up and 1
+1.
7. In a function, the domain is the set of all possible input values which produce real output
values.
Since division by zero is undefined, determine what values of x will make the
denominator zero.
x2 - 21x + 98 = 0
(x - 7)(x - 14) = 0
x = 7 and x = 14
So, the function is undefined when x = 7 and x = 14.
Thus, the domain is the set of all real numbers except x = 7 and
x = 14, which can be written as x < 7, or 7 < x < 14, or x > 14.
8. The recursive formula for a geometric sequence is given below, where t0 is first term, r
is the common ratio, and n > 1.
In this sequence, the common ratio is 3.
The example below shows the formula for finding the third term in the sequence.
Therefore, the recursive formula is shown below.
9. Sixth degree functions, like h(x), have a variable rate of change.
Cubic functions, like g(x), have a variable rate of change.
Exponential functions, like f(x), have a variable rate of change.
Therefore, f(x), g(x), and h(x) all have variable rates of change.
10 A function is symmetric about the origin if the graph of the function remains unchanged
. when reflected across both the x- and the y-axes.
The function graphed has this characteristic.
Therefore, the function is symmetric about the origin.
11 First, find the product of the left side of the equation.
.
Since the first matrix is a 2 × 2 matrix and the second matrix is a 2 × 1 matrix, the
product of the two matrices will be a 2 × 1 matrix.
To find the product of two matrices, multiply the elements of each row in the first matrix
by the elements of each column in the second matrix.
Then, add the products.
Next, substitute the product back into the original equation.
Now, write as a system of linear equations.
12 First, simplify the portion of the terms under the square root sign.
.
and
Now, put the results back into the expression, combine like terms, and add.
13 To solve this problem, use the distributive property and collect like terms.
.
-4(a + 9b) - 6b + (4a - 5c) + 13(-4b - 5c)
-4a - 36b - 6b + 4a - 5c - 52b - 65c
(-4a + 4a) + (-36b - 6b - 52b) + (-5c - 65c)
-94b - 70c
14 When multiplying algebraic fractions, it is often helpful to do any factoring and canceling
. before multiplying the fractions.
To multiply the fractions, multiply across the numerators and across the denominators;
it is not necessary to have common denominators.
x2 + 2x + 1
x-1
•
x-2
x+1
=
=
(x + 1)(x + 1)
x-1
•
x-2
x+1
(x + 1)(x - 2)
x-1
15 The polynomial x2 - 7x + 12 is in the form ax2 + bx + c, where a = 1,
. b = -7, and c = 12. Find the factors of a and c.
1: 1
12: 1, 2, 3, 4, 6, 12
Form two pairs of factors from one factor of a and one factor of c, so that when the
factors are multiplied and then added together (because c is positive), they equal the
absolute value of b.
(1 · 4) + (1 · 3) = 7
Use these numbers to form the factors of the polynomial. Because b is negative, both
factors will have subtraction signs.
(x - 4)(x - 3)
Therefore, (x - 4)(x - 3) is the factored equivalent of x2 - 7x + 12.
16 The solutions of a quadratic equation are the x-intercepts of the graph.
.
Since the solutions are x = -1 and x = -2, find the graph whose x-intercepts are -1 and 2.
Graph W is the only graph whose solutions are x = -1 and x = -2.
17 Isolate the variable, then eliminate the radical by squaring both sides of the equation.
.
Now, substitute x = 144 back into the original equation to make sure that it is a valid
solution.
The solution x = 144 satisfies the equation; therefore, it is a valid solution.
18 First, simplify the left side of the equation, and then cross-multiply.
.
Next, write the equation in standard form, factor, and then solve for
.
Then, check for extraneous solutions by substituting the solutions into the original
equation.
Therefore,
= 10 and
= -20.
19 To divide two complex numbers, multiply the dividend (numerator) and divisor
. (denominator) by the conjugate of the divisor. Remember that
.
In this case, the divisor is
, so its conjugate is
.
20 The graph opens up, so the absolute value is not multiplied by a negative. This
. eliminates f(x) = -|3x + 3| + 3 and f(x) = -|3x - 3| + 3 which would open down.
Next, notice that the graph has two x-intercepts at (-2,0) and (0,0).
Find the function where both of these two points make a true statement when
substituted into the equation.
f(x) = |3x + 3| - 3
f(x) = |3x + 3| - 3
0 = |3(-2) + 3| - 3
0 = |3(0) + 3| - 3
0 = |-3| - 3
0 = |3| - 3
0= 0
0= 0
true
true
This is the only remaining function where (-2,0) and (0,0) both make true statements
when substituted into the equation.
Therefore, the correct answer is f(x) = |3x + 3| - 3.
21 The domain of a function is the x-values.
.
This is a piecewise function on an interval that is discontinuous. A graph is said to be
discontinuous when there is a break or a gap on the graph.
While the discontinuity exists, this is still a function because it passes the vertical line
test for every point of the function.
The smallest value of x for this graph is the value x = -5 and the largest value of x for
this graph is the value x = 4. However, while the leftmost value is included in the
domain, shown by the closed circle, the rightmost value is not, shown by an open circle.
Therefore, the domain of the function is from -5 to 4, not including 4.
{x | -5 < x < 4 }
22 Solve the inequality to determine the correct graph.
.
|x + 3| + 9 > 11
|x + 3| > 2
x+3>2
or x + 3 < -2
x > -1
x < -5
The correct graph is shown below.
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
23 When an exponential expression is raised to another exponent, multiply the exponents.
.
When dividing exponential expressions with the same base, subtract the exponents.
24 The function has a horizontal asymptote at y = 0.
.
Therefore, the function y = 7x never crosses the x-axis.
Thus, the function does not have an x-intercept.
25 First, rewrite the inequality so that the bases on both sides of the equation are the
. same, then use the properties of exponents to simplify.
(5-3)x + 30 <(52)x
5-3(x + 30) < 52x
Now that the bases are the same on both sides of the equation, bring down the
exponents and solve for x.
-3(x + 30) < 2x
-3x - 90 < 2x
-90 < 5x
-18 < x
26 The maximum coordinate of a function is the highest point of the graph.
.
The maximum coordinate for y = -x2 + 2x - 3 is the point (1,2).
27 Substitute values from the table into the given equations to find the equation which
. represents the number of people who bought coffee.
Use p = d2 + d + 4.
d=2
p = (2)2 + (2) + 4 = 10
d=5
p = (5)2 + (5) + 4 = 34
d=7
p = (7)2 + (7) + 4 = 60
d = 10
p = (10)2 + (10) + 4 = 114
d = 14
p = (14)2 + (14) + 4 = 214
The equation p = d2 + d + 4 fits all the values from the table; therefore, it represents
the number of people who bought coffee.
28
.
29
The range of any logarithmic function is (.
,
).
30 The graph of f(x) = x2 is U-shaped, contains the point (0,0), and is symmetric about the
. y-axis.
The correct graph is Z.
31 Consider the function y = f(x) with the real number r.
.
y = f(x) + r
shifts the graph of f(x) up r units
y = f(x) - r
shifts the graph of f(x) down r units
y = f(x + r)
shifts the graph of f(x) left r units
y = f(x - r)
shifts the graph of f(x) right r units
The given equation y = x - 15 can be seen as a function shifted down 15 units or to the
right 15 units as shown below.
y = f(x) - 15 = x - 15
or
y = f(x - 15) = (x - 15) = x - 15
Therefore, the only given choice that is correct is the following:
it is the graph of y = x translated 15 units down.
32 Consider the function y = f(x) with the real numbers a > 0, h > 0, and r > 0.
.
y = f(x) + r
shifts the graph of f(x) up r units
y = f(x) - r
shifts the graph of f(x) down r units
y = f(x + h)
shifts the graph of f(x) left h units
y = f(x - h)
shifts the graph of f(x) right h units
y = a·f(x), where a > 1
stretches the graph of f(x) vertically by a factor of a
y = a·f(x), where 0 < a < 1
compresses the graph of f(x) vertically by a factor of a
The given equation y = 4(x - 7)2 - 8 can be seen as y = f(x) = x2 undergoing three
transformations.
y = 4f(x - 7) - 8 = 4(x - 7)2 - 8
Therefore, the given equation represents the graph of y = x2 vertically stretched,
and then translated 8 units down and 7 units to the right.
33 Notice that g(x) = f(x + 9). Since the change happens inside the function, we know it
. must be a horizontal shift. Since the number is being added to x, the graph of g(x) was
created by shifting the graph of f(x) 9 units to the left.
34 Consider a function r(x) with the real number k.
.
k·r(x) with k > 1
k·r(x) with 0 < k < 1
stretches the graph of r(x) vertically
shrinks the graph of r(x) vertically
The graph of g(x) is a shrunken version of f(x), so either
g(x) = (1/2)|x| or g(x) = (1/3)|x|. Testing some points reveals that
g(x) = (1/3)|x|.
Therefore, g(x) = (1/3)|x| .
35 Since a horizontal shift means a change will occur inside the function, to find the
. equation that would represent the graph of f(x) shifted 3 units to the left, find f(x + 3).
g(x) = f(x + 3) =
(x + 3)
36 Notice that g(x) = f(x - 3). Since the change happens inside the function, we know it
. must be a horizontal shift. Since the number is being subtracted from x, the graph of
g(x) was created by shifting the graph of f(x) 3 units to the right.
37 The range of a function is the set of all output values, typically referred to as the y
. values.
In the graph above, the output values are 0 and all real numbers greater than 0.
Therefore, the range of the function above is [0,
).
38 The first term of this arithmetic sequence is u(1) = 9.
.
Find the common difference in each of the numbers of the sequence.
21 - 18 = 3
18 - 15 = 3
15 - 12 = 3
12 - 9 = 3
Each term in the sequence has increased by 3.
Subtract 3 from the initial number in the sequence to determine the constant term in the
function.
9-3=6
Multiply the common difference of 3 by the number in the sequence, n, and add it to 6.
n
u(n)
Formula
Term
1
u(1)
6 + 1(3)
9
2
u(2)
6 + 2(3)
12
3
u(3)
6 + 3(3)
15
4
u(4)
6 + 4(3)
18
n
u(n)
9 + n(3)
6 + n(3)
Now, write this as an explicit formula.
u(n) = 6 + 3n, and n = 1, 2, 3, 4, . . .
39 Use the given information to set up a linear equation for each family's trip.
.
The total cost of the trip is the dependent variable.
The number of nights for lodging is the independent variable.
The airfare is part of the total cost no matter how many nights they stay; it is the yintercept.
The price per night for lodging is the rate of change, or slope, of the equation.
Set up the equation for each family's trip, solve for the cost per night of lodging, and
then find the difference between those costs.
Johnsons:
$5,100 = 6(cost per night of lodging) + (7)($300)
$5,100 = 6(cost per night of lodging) + $2,100
$3,000 = 6(cost per night of lodging)
$500 = cost per night of lodging
Thompsons:
$1,800 = 6(cost per night of lodging) + (4)($300)
$1,800 = 6(cost per night of lodging) + $1,200
$600 = 6(cost per night of lodging)
$100 = cost per night of lodging
The Johnsons spent $400 more per night.
40 If f(x) is symmetric about the y-axis, then f(-x) = f(x).
.
If f(x) is symmetric about the origin, then f(-x) = -f(x).
Determine the symmetry of f(x) by calculating f(-x).
Therefore, the function is symmetric about the origin because f(-x) = -f(x).
41 Consider the linear system below.
.
Set the matrices A, X, and B equal to the following.
Using matrix multiplication, the system can be written as the matrix equation AX = B.
42 First, simplify the portion of the terms under the radical.
.
Now, put the results back into the expression, combine like terms, and multiply.
43 Use the FOIL method (First Outer Inner Last) to multiply the two expressions. Then
. combine like terms.
(2x + 4)(x - 4) = (2x)(x) + (-4)(2x) + (4)(x) + (4)(-4)
= 2x2 - 8x + 4x - 16
= 2x2 + (-8x + 4x) - 16
= 2x2 - 4x - 16
44 To add algebraic fractions with different denominators, use the least common multiple to
. create a common denominator and simplify.
45 The polynomial x2 - 8x + 15 is in the form ax2 + bx + c, where a = 1,
. b = -8, and c = 15. Find the factors of a and c.
1: 1
15: 1, 3, 5, 15
Form two pairs of factors from one factor of a and one factor of c, so that when the
factors are multiplied and then added together (because c is positive), they equal the
absolute value of b.
(1 · 5) + (1 · 3) = 8
Use these numbers to form the factors of the polynomial. Because b is negative, both
factors will have subtraction signs.
(x - 5)(x - 3)
Therefore, (x - 5)(x - 3) is the factored equivalent of x2 - 8x + 15.
46 The solutions of 0 = -16t2 + 92t + 3 are where the graph of
. y = -16t2 + 92t + 3 crosses the x-axis.
The graph of y = -16t2 + 92t + 3 crosses the x-axis at approximately
x = 0 and x = 5.8.
47 Isolate the variable, and then eliminate the radical by squaring both sides of the
. equation.
Now, substitute x = 49 back into the original equation to make sure it is a valid solution.
The solution x = 49 satisfies the equation; therefore, it is a valid solution.
48 First, combine the two fractions by finding a common denominator.
.
Next, cross-multiply, and then solve for
.
Then, check for extraneous solutions by substituting the solution into the original
equation.
Therefore,
= 33.
49 The complex conjugate of a complex number a + bi is defined to be a - bi. According to
. this, the complex conjugate is -6 + 9i.
50 The tables all have the same x-values: -3, -2, -1, 0, and 1.
.
The points on the graph with these x-values are shown below.
(-3,1), (-2,2), (-1,3), (0,2), (1,1)
Therefore, the following table corresponds to the graph.
x
-3
-2
-1
0
1
y
1
2
3
2
1
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