1. Graphs of Quadratic Functions Day 2

2. The axis of symmetry is the vertical line passing through the vertex
1. Find the equation for the axis of symmetry just from symmetric points:
(3, 10) and (15, 10)
Since these ordered pairs are points of symmetry (same y-value, height on graph), you can find the axis of symmetry by finding the middle.
The middle of 3 and 15 is 9.
The axis of symmetry would be x = 9.
2. Find the equation for the axis of symmetry from formula:
y = ax2+bx+c

3. x = -1 is the axis of symmetry
Graph y = - 𝟏 𝟐 x2 - x + 4
1. Find the axis of symmetry.
y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4
x = −𝑏 2𝑎 = −(−1) 2(−1/2) = 1 −1 = -1
x = -1 is the axis of symmetry
2. Find the vertex.
plug in x = -1
y = - 𝟏 𝟐 (-1)2 – (-1) y=4.5
Vertex = (-1,4.5)

4. y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4
Graph y = - 𝟏 𝟐 x2 - x + 4 continued
3. Graph 2 more points
Try the y-intercept (c value)
y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4
y-intercept (0,4)
Find more points
Select any x value you want and plug into function to find y value (make easy choices, ie. whole numbers
Select x=1, and plug in
y = - 𝟏 𝟐 (1)2 – (1) + 4
y = - 𝟏 𝟐 y=2.5 makes point (1,2.5)

5. Graph y = - 𝟏 𝟐 x2 - x + 4 continued
4. Graph all points and mirror images to make symmetric parabola
axis of symmetry x=-1
Vertex (-1,4.5)
(0,4) and mirror image
(-2,4)
(1,2.5) and mirror image
(-3,2.5)
Check: Opens down because a is negative

6. Graph Quadratic Example Notes:

7. x = 𝟏 𝟐 is the axis of symmetry
Graph y = x2-x-6
1. Find the axis of symmetry.
y = ax2+bx+c a=1, b=-1, and c=-6
x = −𝑏 2𝑎 = −(−1) 2(1) = 1 2
x = 𝟏 𝟐 is the axis of symmetry
2. Find the vertex.
plug in x = 𝟏 𝟐
y = ( 𝟏 𝟐 )2 – ( 𝟏 𝟐 ) y = -6 𝟏 𝟒
Vertex = ( 𝟏 𝟐 , -6 𝟏 𝟒 )

8. y = ax2+bx+c a=1, b=-1, and c=-6
Graph y = x2-x-6 continued
3. Graph at least 2 more points
Try the y-intercept (c value)
y = ax2+bx+c a=1, b=-1, and c=-6
y-intercept (0,-6)
Find more points
Select any x value you want and plug into function to find y value (make easy choices, ie. whole numbers
Select x=2, and plug in
Select x=3, and plug in
y = (𝟐)2 – (𝟐) - 6
y=-4 makes point (2,-4)
y = (𝟑)2 – (𝟑) - 6
y=0 makes point (3,0)

9. Graph y = x2-x-6 continued
4. Graph all points and mirror images to make symmetric parabola
axis of symmetry x = 𝟏 𝟐
Vertex ( 𝟏 𝟐 , -6 𝟏 𝟒 )
(0,-6) and mirror image
(1,-6)
(2,-4) and mirror image
(-1,-4)
(3,0) and mirror image
(-2,0)
Check: Opens up because a is positive

10. Additional Practice Problems if needed

11. a is negative: opens down
Graph: y= -2x2+2x+1
a is negative: opens down
Find Line of Symmetry
= 1 2
Find the y value, then pick more points to see how to draw the parabola.

12. 2 - - 2 - 2 - 2

13. y=-2x2+2x+1

14. Graph: y= x2+5x-14 Plug back in for y and solve
Will open up b/c a is positive
Find axis of symmetry and the vertex
Plug
back in for y and solve

15. Change to common denominators
Vertex is

16. Vertex is Can also find x intercepts of
y = x2 + 5x by factoring to find solutions
(2,0) and (-7,0)
Vertex is y -7 x 2