1. Graphs of Quadratic Functions Day 1

2. Graph Quadratic Functions:
Standard Form: y = ax2 + bx + c
Shape: Parabola
When in standard form,
If a is positive, the parabola opens up y = ax2+bx+c
If a is negative, the parabola opens down y = -ax2+bx+c

3. Will It Open Up or Down? y = 4x2 + 7x – 4 y = -6.5x2 + 9
*must be in standard form

4. Parts of Parabolas: Axis of Symmetry Axis of Symmetry:
Vertex:
Highest or lowest point of the graph
(the max or min of the function)
Lies on the axis of symmetry
Axis of Symmetry
Axis of Symmetry:
Vertical line of symmetry that divides parabola into two parts that are mirror images.
ALL parabolas are Symmetric!
Calculate with formula
Vertex

5. Graphing Quadratic Functions Steps
(make sure to identify the a, b, and c values) 1. Find the equation of the axis of symmetry & draw the vertical line on the graph 2. Find the vertex coordinates & plot vertex on axis of symmetry (plug x= value of axis of sym. into function and evaluate for y) 3. Find and plot at least 2 more points and their symmetric points (mirror image the same distance across axis of symmetry) -Use the y-intercept (c value) if possible -Pick another ‘easy but logical’ point (select any logical x value, plug in and evaluate for y) 4. Sketch the curve and reflect it across the axis of symmetry

6. x=0 is the axis of symmetry
Graph y = x2
1. Find the axis of symmetry.
y = ax2+bx+c a=1, b=0, and c=0
x=0 is the axis of symmetry
2. Find the vertex.
x=0 is also the x value of the vertex, now find the y value. If x = 0, plug in y = (0)2 y = 0
Vertex = (0,0)

7. Graph y = x2 continued 3. Graph 2 more points
Try the y-intercept (c value)
y = ax2+bx+c a=1, b=0, and c=0
y-intercept (0,0) is also the vertex, so we don’t need to plot it again
Find 2 other points and their symmetric point
Select any x value you want and plug into function to find y value (make easy choices, ie. whole numbers)
Select x=1 and x=2
y = (1)2 y = 1 makes point (1,1)
y = (2)2 y = 4 makes point (2,4)

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

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

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

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

12. Graph Quadratic Example Notes: