1. Vertex Form

2. Forms of quadratics Factored form a(x-r1)(x-r2) Standard Form ax2+bx+c
Vertex Form a(x-h)2+k

3. Each form gives you different information!
Factored form a(x-r1)(x-r2)
Tells you direction of opening
Tells you location of x-intercepts (roots)
Standard Form ax2+bx+c
Tells you location of y-intercept
Vertex Form a(x-h)2+k
Tells you direction opening
Tells you the location of the vertex (max or min)

4. Direction of opening
x2 opens up

5. Direction of opening
ax2 stretches x vertically by a
Here a is 1.5

6. Direction of opening ax2 stretches x vertically by a Here a is 0.5
Stretching by a fraction is a squish

7. Direction of opening ax2 stretches x vertically by a Here a is -0.5
Stretching by a negative causes a flip

8. Direction of opening a is the number in front of the x2
The value a tells you what direction the parabola is opening in.
Positive a opens up
Negative a opens down
The a in all three forms is the same number
a(x-r1)(x-r2)
ax2+bx+c
a(x-h)2+k

9. Factored form a(x-r1)(x-r2)
a is the direction of opening
r1 and r2 are the x-intercepts
Or roots, or zeros
Example: -2(x-2)(x+0.5)
a is negative, opens down.
r1 is 2, crosses the x-axis at 2.
r2 is -0.5, crosses the x-axis at -0.5

10. Factored form a(x-r1)(x-r2)
a is the direction of opening
r1 and r2 are the x-intercepts
Or roots, or zeros
Example: -2(x-2)(x+0.5)
a is negative, opens down.
r1 is 2, crosses the x-axis at 2.
r2 is -0.5, crosses the x-axis at -0.5

11. Standard form ax2+bx+c a is the direction of opening
c is the y-intercept
ƒ(0)=a02+b0+c=c
Example: -2x2+3x+2
Opens down
Crosses through the point (0,2)

12. Standard form ax2+bx+c a is the direction of opening
c is the y-intercept
ƒ(0)=a02+b0+c=c
Example: -2x2+3x+2
Opens down
Crosses through the point (0,2)

13. Vertex form
Start with f(x)=x2

14. Vertex form
Stretch/Flip if you want
aƒ(x)=ax2

15. Vertex form
Shift right by h
aƒ(x-h)=a(x-h)2 h

16. Vertex form
Shift up by k
aƒ(x-h)+k=a(x-h)2+k k h

17. Vertex form
Define a new function
g(x)=a(x-h)2+k
(h,k)

18. Vertex form a(x-h)2+k a tells you direction of opening
(h,k) is the vertex
(h,k)

19. Vertex form a(x-h)2+k a tells you direction of opening
(h,k) is the vertex
Example: -2(x-3/4)2+25/8
Opens down
Has vertex at (3/4, 25/8)

20. Vertex form a(x-h)2+k a tells you direction of opening
(h,k) is the vertex
Example: -2(x-3/4)2+25/8
Opens down
Has vertex at (3/4, 25/8)
(3/4, 25/8)

21. Switching between forms Gives you a full picture
Example:
ƒ(x)=-2(x-2)(x+0.5)
ƒ(x)=-2x2+3x+2
ƒ(x)=-2(x-3/4)2+25/8
are all the same function
Opens down
Crosses x axis at 2 and -0.5
Crosses the y-axis at 2
Has vertex at (3/4, 25/8)

22. Switching between forms Gives you a full picture
Example:
ƒ(x)=-2(x-2)(x+0.5)
ƒ(x)=-2x2+3x+2
ƒ(x)=-2(x-3/4)2+25/8
are all the same function
Opens down
Crosses x axis at 2 and -0.5
Crosses the y-axis at 2
Has vertex at (3/4, 25/8)

23. The graph of f(x) has a negative y-intercept B) f(x) has 2 real zeros.
Consider the function f(x) = -3x2+2x-9. Which of the following are true?
The graph of f(x) has a negative
y-intercept
B) f(x) has 2 real zeros.
C) The graph of f(x) attains a maximum value
D) Both (A) and (B) are true
E) Both (A) and (C) are true.

24. Standard form: ax2+bx+c.
Consider the function f(x) = -3x2+2x-9. Which of the following are true?
Standard form: ax2+bx+c.
a is negative: opens down. ƒ(x) attains a maximum value. (C) is true.
c is my y-intercept. c is negative. My y-intercept is negative. (A) is true.
E) Both (A) and (C) are true.

30. The Vertex Formula
Remember the Quadratic formula

31. What does the QF say?

32. The Vertex Formula

33. Example

34. x = - 9 x = 9 x = 2 x = 6 None of the above.
Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry.
x = - 9
x = 9
x = 2
x = 6
None of the above.

35. The axis of symmetry is halfway between the roots.
Given the function R(x)=(2x+6)(x-12), find an equation for its axis (line) of symmetry.
The roots are x=-3 and x=12.
The axis of symmetry is halfway between the roots.
(12-3)/2=4.5, the number halfway between -3 and 12.
x=4.5 is the axis of symmetry
E) None of the above.

36. How to find an equation from vertex and point
A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola?

37. How to find an equation from vertex and point
A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola?
(h,k)=(1,3)
(x1,y1)=(0,1)

38. How to find an equation from vertex and point
A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola?
(h,k)=(1,3)
(x1,y1)=(0,1)
But to be finished, I need to know a!
Use: My formula is true for every x,y including x1,y1

39. How to find an equation from vertex and point
A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola?
(h,k)=(1,3)
(x1,y1)=(0,1)
My formula is true for every x,y; not just x1,y1

40. y = (x+2)2 y = x2+3 y = x2+1 y = x2 None of the above
A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola.
y = (x+2)2
y = x2+3
y = x2+1
y = x2
None of the above

41. A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. E