Chapter 8.2(a) Characteristics of Quadratic Functions.notebook
February 15, 2017
Bellwork:
Homework Questions?
Solve and graph the following inequalities:
1) x + 4 > 8
2) 4 2x < 24
3) Make a table of values and graph the function y = 3x2.
4) The cost to manufacture x pairs of sunglasses can be represented by a function C(x). If it costs $398 to manufacture 4 pairs of sunglasses, which of the following is true?
A. C(4) = 99.50
B. C(398) = 4
C. C(4) = 398
D. C(99.50) = 1 Feb 10­3:46 PM
Feb 10­3:57 PM
What is the x­intercept of a function? How do you find the x­intercept?
Chapter 8.2 Characteristics of quadratic Functions
Identify the zeros, axis of symmetry and vertex of a parabola. Find the axis of symmetry and vertex of a parabola from a quadratic function.
A zero of a function is an x­value that makes the function equal to 0. So a zero of a function is the same as an x­
intercept of a function.
A Quadratic Function may have one, two or no zeros.
No Zeros
Feb 10­3:58 PM
2) Find the zeros of each quadratic function from its graph. Check.
Two Zeros
Feb 10­3:59 PM
2) Find the zeros of each quadratic function from its graph. Check.
c. y = ­2x2 ­ 2
a. y = x2 ­ 2x ­ 3
One Zero
d. y = x2 ­ 4 b. y = x2 + 8x + 16
Feb 10­4:10 PM
Feb 10­4:10 PM
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Chapter 8.2(a) Characteristics of Quadratic Functions.notebook
A vertical line divides every parabola into two symmetrical halves. This line is called the Axis of Symmetry. The axis of symmetry always passes through the vertex of the parabola.
February 15, 2017
3) Find the axis of symmetry of each parabola:
We can find the axis of symmetry by using the zeros of a quadratic function.
a.
b.
Finding the Axis of Symmetry
Words
Numbers
Graph
One Zero:
Vertex (3, 0)
If a function has one zero, use the x­coordinate of the vertex to find the axis Axis of Symmetry x = 3
of symmetry.
(­4 + 0)/2 Two Zeros:
= ­4/2 = ­2
If a function has two zeros, use the average of the two zeros to find the Axis of Symmetry axis of symmetry.
x = ­2
Feb 10­4:19 PM
3) Find the axis of symmetry of each parabola:
c.
d. Feb 10­4:30 PM
Before we can find the axis of symmetry we need to be able to identify the parts of a quadratic function in standard form.
Standard Form: y = ax2 + bx + c or f(x) = ax2 + bx + c
1) Identify the a, b, and c values for the following quadratic functions. a. y = 3x2 + 4x ­ 5
b. f(x) = ­2x2 ­ 5x + 5
c. g(x) = 5x + 6 + x2
d. y = 7 ­ 8x2 ­ 4
Feb 10­4:30 PM
We have to use a formula to find the axis of symmetry when a function has no zeros. The formula will work for all quadratics.
Feb 10­12:39 PM
Find the axis of symmetry of the graph.
3x2 ­ 6x + 7 Axis of symmetry by Using the Formula
For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line:
Example: y = 3x2 + 12x + 5
Feb 10­12:39 PM
Feb 10­12:40 PM
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Chapter 8.2(a) Characteristics of Quadratic Functions.notebook
2) Find the axis of symmetry of the graphs.
February 15, 2017
Since the vertex of a parabola always lies on the axis of symmetry we can find the vertex of any parabola.
a. y = ­3x2 + 10x + 9
Finding the vertex of a Parabola:
Step 1: Find the axis of symmetry by using zeros or the formula.
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b. y = 4x ­ 3x + 2
Step 2: To find the corresponding y­coordinate, substitute the x­
value of the axis of symmetry into the function.
c. y = ­2x2 + 6x ­ 5
Feb 10­12:41 PM
3) Find the vertex of the following parabola.
a. y = ­2x + 12x + 5
Step 3: Write the vertex as an ordered pair.
Feb 10­12:41 PM
3) Find the vertex of the following parabolas.
a. f(x) = ­3x2 + 6x ­ 7
b. y = ­2x2 + 8x ­ 9
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c. y = 0.25x2 + 2x + 3
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Application:
The graph f(x) = ­0.06x2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where x represents the horizontal distance in meters form where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain.
Feb 10­12:41 PM
Homework:
P. 535 #1­31 (odds), 18, 36 Feb 10­4:48 PM
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