Download Standard Form of a Quadratic Function

Standard Form of a
Quadratic Function
4-2
Vocabulary
Review
1. Circle the functions in standard form.
y 5 2x2 2 4x 1 2
y 5 13 (x 2 4)
y 5 24x 1 1
5
y 5 232 x 2 3 x 1 4
Write each equation in standard form.
2. x 1 2y 5 17
3. 2x 5 5
y 5 212 x 1 17
2
4. 5 2 x 5 y 1 2
2x 2 5 5 0
y 5 2x 1 3
Vocabulary Builder
Standard Form of a
Quadratic Function
quadratic (adjective) kwah DRAT ik
y ax2 bx c,
Related Words: parabola, vertex, axis of symmetry
Examples: quadratic functions, y 5 x2 , y 5 23x2 1 7, f (x) 5 2x2 1 5x 2 4,
g(x) 5 12 (x 2 4)2 1 5
Nonexamples: not quadratic functions, y 5
2x2
x2 1 5x 1 10
1
,
3x
1 4x 1 5
Use Your Vocabulary
5. Circle the graphs of quadratic functions.
y
y
y
y
x
x
x
x
Chapter 4
86
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a0
Definition: A quadratic function is a function that can be
written in the form y 5 ax2 1 bx 1 c where a 2 0. The
graph of a quadratic function is a parabola.
Problem 1 Finding the Features of a Quadratic Function
Got It? What are the vertex, axis of symmetry, maximum and minimum values,
and range of y 5 23x2 2 4x 1 6?
6. Circle the graph of y 5 23x2 2 4x 1 6.
maximum,
+ 23 , 7 13 ,
axis of
symmetry,
x
฀ 2
3
7. Draw and label the axis of symmetry on the graph you circled in Exercise 6.
8. Circle and label the maximum or minimum value on the graph.
9. Circle the range of the function.
y $ 5.0
y # 6.0
all real numbers # 7.3
all real numbers # 9.2
Properties Quadratic Function in Standard Form
• The graph of f (x) 5 ax2 1 bx 1 c, a 2 0, is a parabola.
• If a . 0, the parabola opens upward. If a , 0, the parabola opens downward.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
b
.
• The axis of symmetry is the line x 5 22a
b
b
• The x-coordinate of the vertex is 22a
. The y-coordinate of the vertex is f Q 22a
R.
• The y-intercept is (0, c).
10. The y-intercept of the graph of f (x) 5 5x2 2 3x 2 4 is Q 0 , 24 R .
Problem 2 Graphing a Function of the Form y 5 ax2 1 bx 1 c
Got It? What is the graph of y 5 22x 2 1 2x 2 5?
b
11. The axis of symmetry is x 5 22a 5 2
y
2
5
2 ? 22
1
2
8
.
4
12. Substitute to find the y-coordinate of the vertex.
x
2
f Q 12 R 5 22 Q 12 R 1 2 Q 12 R 2 5 5 2412
4
2
O
13. The vertex is Q 2 , 242 R
4
14. The y-intercept is 25 . The reflection of the y-intercept across
8
1
1
2
4
the axis of symmetry is Q 1 , 25 R .
15. Plot the points from Exercises 13 and 14. Draw a smooth curve.
87
Lesson 4-2
Problem 3
Converting Standard Form to Vertex Form
Got It? What is the vertex form of y 5 2x2 1 4x 2 5?
16. Use the justifications at the right to find the vertex.
y 5 Q 21 R x2 1 Q 4 R x 1 Q 25 R
Write the function in the form y 5 ax2 1 bx 1 c.
4
b
x 5 22a 5 2
5 2
Find the x-coordinate of the vertex.
2 ? 21
y 5 21(4) 1 4(2) 2 5
Substitute the x-coordinate value into the equation and simplify.
y 5 21
17. The vertex is Q 2 , 21 R .
18. Use the general form of the equation, y 5 a(x 2 h)2 1 k. Substitute for a, h, and k.
2
y 5 21 B x 2 Q 2 R R 1 Q 21 R
y 5 2(x 2 2)2 2 1
19. The vertex form of the function is
Problem 4
.
Interpreting a Quadratic Graph
Got It? The Zhaozhou Bridge in China is the oldest known arch bridge, dating to 605 a.d.
You can model the support arch with the function f (x) 5 20.001075x2 1 0.131148x,
where x and y are measured in feet. How high is the arch above its supports?
Answers may vary. Sample: the middle, the vertex, the maximum
_______________________________________________________________________
21. Find the x-coordinate of the vertex.
0.131148
b
x 5 22a 5 2
5 61
2?
20.001075
22. The axis of symmetry of the parabola is x 5 61 .
23. The length of the bridge is
122
ft.
24. Use the x-coordinate of the vertex to find the y-coordinate.
y 5 20.001075(61)2 1 0.131148(61)
5 4.0000
25. The vertex is about Q 61 , 4 R , so the arch is 4 feet above its support.
Chapter 4
88
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20. What point on the parabola gives the height of the arch above its supports?
Lesson Check • Do you UNDERSTAND?
Error Analysis A student graphed the function
y 5 2x2 2 4x 2 3. Find and correct the error.
x = -4 = -1
26. The vertex of y 5 ax2 1 bx 1 c is Q22a , f Q22a RR .
Find the x- and y-coordinates of the vertex of
b
b
y 5 2x2 2 4x 2 3.
x-coordinate:
2224
? 4
6
2(2)
y = 2(-1)2 - 4(-1) - 3
=2+4-3
=3
vertex (-1, 3)
(1, 3) 2
4 2
0
2
51
y-coordinate:
2(1)2
2 4(1) 2 3 5 25
27. Find the y-intercept of y 5 2x2 2 4x 2 3.
2(0)2 2 4(0) 2 3 5 23
28. Describe the student’s error and graph the function correctly.
y
Answers may vary. Sample: The student miscalculated the
________________________________________________________
4
x-coordinate of the vertex.
________________________________________________________
2
x
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2
O
2
4
2
4
Math Success
Check off the vocabulary words that you understand.
quadratic
standard form
vertex
axis of symmetry
y-intercept
Rate how well you can graph quadratic functions written in standard form.
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Lesson 4-2