1. Grade 8 Algebra I Characteristics of Quadratic Functions
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2. Tell whether each function is quadratic. Explain.
Warm Up
Tell whether each function is quadratic. Explain.
1) y + x = 2x2
2) y = -3x + 20
3) (-2, 4) , (-1, 1) , (0, 0) , (1, 1) , (2, 4)
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3. Characteristics of Quadratic Functions
An x-intercept of a function is a value of x when y = 0.
A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function.
Since a graph intersects the x-axis at the point or points containing an x-intercept these intersections are also at the zeros of the function.
A quadratic function may have one, two, or no zeros.
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4. Finding Zeros of Quadratic Functions From Graphs
Find the zeros of each quadratic function from its graph. Check your answer.
A) y = x2 - x - 2 -2 2 x y 4
The zeros appear to be -1 and 2.
Check:
y = x2 - x – 2
y = (-1)2 – (-1) - 2
= = 0
y = (2)2 – (2) – 2
= = 0
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5. The only zero appears to be 1.
B) y= -2x2 + 4x - 2
The only zero appears to be 1. -2 2 x y -4
Check:
y = -2x2 + 4x - 2
y = -2(1)2 + 4(1) - 2
= = 0
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6. C) y = x2 + 1 4 -2 2 x y 4
The graph does not cross the x-axis, so there are no zeros of this function.
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7. Now you try!
Find the zeros of each quadratic function from its graph. Check your answer.
1) y=-4x2 - 2
2) y= x2 - 6x + 9 -2 2 x y -4 -6 y 4 2 x 6
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8. Finding the Axis of Symmetry by Using Zeros
A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry.
The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry.
One Zero
If a function has one zero, use the x-coordinate of the
vertex to find the axis of symmetry. y 4 2 x 6
Vertex: (3, 0)
Axis of symmetry: x = 3
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9. If a function has two zeros, use the average of the two
zeros to find the axis of symmetry. -2 2 x y -4 -6 4
-4, 0
0, 0
x = -2
= -4 = -2
Axis of symmetry: x = -2
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10. Finding the Axis of Symmetry by Using Zeros
Find the axis of symmetry of each parabola. -2 2 x y -4 -6 4 A)
Identify the x-coordinate of the vertex.
(2, 0)
The axis of symmetry is x = 2.
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11. Find the average of the zeros.y B)
1 + 5 = 6 = 3 2 x
Find the average of the zeros. -2 2 4 6 -4 -2 -4 -6 -8
The axis of symmetry is x = 3.
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12. Find the axis of symmetry of each parabola.
Now you try!
Find the axis of symmetry of each parabola. A) B)
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13. Finding the Axis of Symmetry by Using the Formula
If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions.
FORMULA
For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line
x = - b 2a
EXAMPLE
y = 2x2 + 4x + 5
x = - b 2a
x = = -1
2(2)
The axis of symmetry is x = -1.
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14. Finding the Axis of Symmetry by Using the Formula
Find the axis of symmetry of the graph of y = x2 + 3x + 4.
Step1 Find the values of a and b.
y = 1x2 + 3x + 4
a = 1, b = 3
Step2 Use the formula x = - b 2a
x = = - 3 = -1.5
2(1)
The axis of symmetry is x = -1.5.
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15. 1) Find the axis of symmetry of the graph of y = 2x2 + x + 3.
Now you try!
1) Find the axis of symmetry of the graph of y = 2x2 + x + 3.
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16. Finding the Vertex of a Parabola
Once you have found the axis of symmetry, you can use it to identify the vertex.
Step 1: To find the x-coordinate of the vertex, find the axis of symmetry by using zeros or the formula.
Step 2: To find the corresponding y-coordinate, substitute the x-coordinate of the vertex into the function.
Step 3: Write the vertex as an ordered pair.
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17. Finding the Vertex of a Parabola
Find the vertex.
A) y = - x2 - 2x
Step 1: Find the x-coordinate.
The zeros are -2 and 0.
X = = -2 = -1
Step 2: Find the corresponding y-coordinate.
y = - x2 - 2x
Use the function rule.
= - (-1)2 - 2(-1) = 1
Substitute -1 for x.
Step 3: Write the ordered pair.
(-1, 1)
The vertex is (-1, 1) .
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18. Step 1: Find the x-coordinate.
B) y = 5x2 - 10x + 3
Step 1: Find the x-coordinate.
a = 5, b = -10
Identify a and b.
x = b = - (- 10) = 10 = 1
2a (5)
Substitute 5 for a and -10 for b.
Step 2: Find the corresponding y-coordinate.
y = 5x2 - 10x + 3
Use the function rule.
= 5(1) (1) + 3
= = -2
Substitute 1 for x.
Step 3: Write the ordered pair.
(1, -2)
The vertex is (1, -2) .
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19. 1) Find the vertex of the graph of y = x2 - 4x - 10.
Now you try!
1) Find the vertex of the graph of y = x2 - 4x - 10.
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20. Architecture Application
The height above water level of a curved arch support for a bridge can be modeled by f (x) = x x + 0.8, where x is the distance in feet from where the arch support enters the water. Can a sailboat that is 24 feet tall pass under the bridge? Explain.
The vertex represents the highest point of the arch support.
Step 1: Find the x-coordinate.
a = , b = 0.84
Identify a and b.
x = -b = - (0.84) = 60
2a 2(-0.007)
Substitute for a and 0.84 for b.
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21. Step 2: Find the corresponding y-coordinate.
f (x) = x x + 0.8
Identify a and b.
= (60) (60) + 0.8
= 26
Substitute 60 for x.
Since the height of the arch support is 26 feet, the sailboat can pass under the bridge.
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22. Now you try!
1) The height of a small rise in a roller coaster track is modeled by f (x) = -0.07x x , where x is the distance in feet from a support pole at ground level. Find the height of the rise.
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23. Find the zeros of each quadratic function from its graph.
Assessment
Find the zeros of each quadratic function from its graph. 2) 1)
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24. Find the axis of symmetry of each parabola.3) 4)
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25. For each quadratic function, find the axis of symmetry of its graph.
5) y = x2 + 4x - 7
6) y = 3x2 - 18x + 1
7) y = 2x2 + 3x - 4
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26. Find the vertex of each parabola.
8) y = -5x2 + 10x + 3
9) y = x2 + 4x - 7
10) The height in feet above the ground of an arrow after it is shot can be modeled by y = -16t2 + 63t + 4. Can the arrow pass over a tree that is 68 feet tall? Explain.
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27. Characteristics of Quadratic Functions
Let’s review
Characteristics of Quadratic Functions
An x-intercept of a function is a value of x when y = 0.
A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function.
Since a graph intersects the x-axis at the point or points containing an x-intercept these intersections are also at the zeros of the function.
A quadratic function may have one, two, or no zeros.
CONFIDENTIAL

28. Finding Zeros of Quadratic Functions From Graphs
Find the zeros of each quadratic function from its graph. Check your answer.
A) y = x2 - x - 2 -2 2 x y 4
The zeros appear to be -1 and 2.
Check:
y = x2 - x – 2
y = (-1)2 – (-1) - 2
= = 0
y = (2)2 – (2) – 2
= = 0
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29. C) y = x2 + 1 4 -2 2 x y 4
The graph does not cross the x-axis, so there are no zeros of this function.
CONFIDENTIAL

30. Finding the Axis of Symmetry by Using Zeros
A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry.
The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry.
One Zero
If a function has one zero, use the x-coordinate of the
vertex to find the axis of symmetry. y 4 2 x 6
Vertex: (3, 0)
Axis of symmetry: x = 3
CONFIDENTIAL

31. If a function has two zeros, use the average of the two
zeros to find the axis of symmetry. -2 2 x y -4 -6 4
-4, 0
0, 0
x = -2
= -4 = -2
Axis of symmetry: x = -2
CONFIDENTIAL

32. Finding the Axis of Symmetry by Using the Formula
If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions.
FORMULA
For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line
x = - b 2a
EXAMPLE
y = 2x2 + 4x + 5
x = - b 2a
x = = -1
2(2)
The axis of symmetry is x = -1.
CONFIDENTIAL

33. Finding the Axis of Symmetry by Using the Formula
Find the axis of symmetry of the graph of y = x2 + 3x + 4.
Step1 Find the values of a and b.
y = 1x2 + 3x + 4
a = 1, b = 3
Step2 Use the formula x = - b 2a
x = = - 3 = -1.5
2(1)
The axis of symmetry is x = -1.5.
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34. You did a great job today!
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