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Topic 7 & 8
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2. Students Will: Graph and transform quadratic functions.
Quadratic Functions Part: Section Topics 7 & 8
Students Will:
Graph and transform quadratic functions.

3. x = 0; (0, –4); opens downward
Quadratic Functions Part: Section Topics 7 & 8
Let’s Recall
For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward.
1. y = x2 + 3
2. y = 2x2
3. y = –0.5x2 – 4
x = 0; (0, 3); opens upward
x = 0; (0, 0); opens upward
x = 0; (0, –4); opens downward

4. For the parent function f(x) = x2:
Quadratic Functions Part: Section Topics 7 & 8
The quadratic parent function is f(x) = x2. The graph of all other quadratic functions are transformations of the graph of f(x) = x2.
For the parent function f(x) = x2:
The axis of symmetry is x = 0, or the y-axis.
The vertex is (0, 0)
The function has only one zero, 0.

5. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can move it up or down by adding a constant to the y-value:
Note: to move the line down, we use a negative value for C.
C > 0 moves it up
C < 0 moves it down

6. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can move it left or right by adding a constant to the x-value:
Subtracting C moves the function to the right (the positive direction)
C > 0 moves it up
C < 0 moves it down

7. add to y, go high add to x, go left
Quadratic Functions Part: Section Topics 7 & 8
A way to remember what happens to the graph when we add a constant:
add to y, go high add to x, go left

8. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can stretch or compress it in the y-direction by multiplying the whole function bya constant
C > 1 stretches it
0 < C < 1 compresses

9. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can stretch or compress it in the x-direction by multiplying x by a constant.
C > 1 compresses it
0 < C < 1 stretches it

10. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can stretch or compress it in the y-direction by multiplying the whole function bya constant
C > 1 stretches it
0 < C < 1 compresses

11. Quadratic Functions Part: Section Topics 7 & 8
 This is the quadratic parent function f(x) = x2
We can flip it upside down by multiplying the whole function by −1:
This is also called: 
reflection about the x-axis
We can combine a negative value with a scaling:

12. Summary Quadratic Functions Part: Section Topics 7 & 8 y = f(x) + C
C > 0 moves it up
C < 0 moves it down
y = f(x + C)
C > 0 moves it left
C < 0 moves it right
y = Cf(x)
C > 1 stretches it in the y-direction
0 < C < 1 compresses it
y = f(Cx)
C > 1 compresses it in the x-direction
0 < C < 1 stretches it
y = −f(x)
Reflects it about x-axis
y = f(−x)
Reflects it about y-axis

13. Summary with Examples f(x) = x2 + 2 f(x) = x2 − 3 f(x) = (x−4)2
Quadratic Functions Part: Section Topics 7 & 8
Summary with Examples
Move 2 spaces up:
f(x) = x2 + 2
Move 3 spaces down:
f(x) = x2 − 3
Move 4 spaces to the right:
f(x) = (x−4)2 
Move 5 spaces to the left:
f(x) = (x+5)2  
Stretch it by 2 in the y-direction:
f(x) = 2x2 
Compress it by 3 in the x-direction:
f(x) = (3x)2 
Flip it upside down:
f(x) = −x2 

14. Quadratic Functions Part: Section Topics 7 & 8

15. Quadratic Functions Part: Section Topics 7 & 8
The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

16. Order the functions from narrowest graph to widest.
Quadratic Functions Part: Section Topics 7 & 8
Order the functions from narrowest graph to widest.
f(x) = 3x2, g(x) = 0.5x2
Step 1 Find |a| for each function.
|3| = 3
|0.05| = 0.05
Step 2 Order the functions.
f(x) = 3x2
g(x) = 0.5x2
The function with the narrowest graph has the greatest |a|.

17. Example 1B: Comparing Widths of Parabolas
Quadratic Functions Part: Section Topics 7 & 8
Example 1B: Comparing Widths of Parabolas
Order the functions from narrowest graph to widest.
f(x) = x2, g(x) = x2, h(x) = –2x2
Step 1 Find |a| for each function.
|1| = 1
|–2| = 2
Step 2 Order the functions.
h(x) = –2x2
The function with the narrowest graph has the greatest |a|.
f(x) = x2
g(x) = x2

18. Quadratic Functions Part: Section Topics 7 & 8

19. Quadratic Functions Part: Section Topics 7 & 8
The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of f(x) = ax2 up or down the y-axis.

20. Quadratic Functions Part: Section Topics 7 & 8

21. Compare the graph of the function with the graph of f(x) = x2
Quadratic Functions Part: Section Topics 7 & 8
Example 2A Continued
Compare the graph of the function with the graph of f(x) = x2
g(x) = x2 + 3
The vertex of
f(x) = x2 is (0, 0).
g(x) = x2 + 3
is translated 3 units up to (0, 3).
The axis of symmetry is the same.

22. Compare the graph of each the graph of f(x) = x2.
Quadratic Functions Part: Section Topics 7 & 8
Compare the graph of each the graph of f(x) = x2.
g(x) = –x2 – 4
Method 1 Compare the graphs.
The graph of g(x) = –x2 – 4
opens downward and the graph
of f(x) = x2 opens upward.
The axis of symmetry is the same.
The vertex of g(x) = –x2 – 4
f(x) = x2 is (0, 0).
is translated 4 units down to (0, –4).
The vertex of

23. Compare the graph of the function with the graph of f(x) = x2.
Quadratic Functions Part: Section Topics 7 & 8
Compare the graph of the function with the graph of f(x) = x2.
g(x) = x2 + 2
Method 1 Compare the graphs.
The graph of
g(x) = x2 + 2
is wider than the graph of
f(x) = x2.
The graph of
g(x) = x2 + 2
opens upward and the graph
of f(x) = x2 opens upward.

24. Graphing Vertex Form:

25. Homework: Study Guide & Khan Academy