1. Analyze Graphs of Quadratic Functions

2. Quadratic Graphs Vocabulary
End Behavior of a Graph:
Describes if a function opens up or down and will have a min. or max.
If a is positive (a > 0), then
f(x) opens up
has a minimum value (at the y-coordinate of the vertex, like range)
If a is negative (a < 0), then
f(x) opens down
has a maximum value (at the y-coordinate of the vertex, like range)
To find the min/max value, find vertex using
The y-coordinate of vertex is the function’s min/max value

3. Quadratic Graphs Vocabulary
x-intercepts = Where the function crosses the x axis.
There could be 2, 1, or no intercepts.
The x-intercepts can also be called “roots of the function”, “zeros of the function”, “solutions to the function”
To find the x-intercepts, solve the quadratic. The x-intercepts are the solutions
Solve by factoring or any other method you like

4. Find the x-intercepts / roots and the min or max value
Example 2:
y = x2 - 2x - 3
Example 1:
y = x2 + 8x
Find the x-intercept/roots by solving the quadratic
y=(x)(x+8)
0=(x)(x+8)
x=0 x=-8
x-intercepts / roots:
(0,0) and (0,-8)
y=(x-3)(x+1)
0=(x-3)(x+1)
x=3 x=-1
x-intercepts / roots:
(0,3) and (0,-1)
Find the vertex in order to identify the min/max of the function
x= −𝑏 2𝑎 = −8 2(1) = -4
y=(-4)2 + 4(-4)
y=-16 vertex (-4,-16)
Min. value is -16
x= −𝑏 2𝑎 = −(−2) 2(1) = 1
y=(1)2 - 2(1) - 3
y=-4 vertex (1,-4)
Min. value is -4

5. Transformations on Quadratic Graphs
Translate = slide or shift
adding/subtracting to the y value shifts up and down (it adds onto the end of the function and changes the y-int)
c(x)=x is 3 units higher than g(x)=x2
d(x)=x is 3 units lower than g(x)=x2
adding/subtracting to the x value shifts right and left (it affects the zeros)
k(x)=(x - 3)2 is 3 units to the right of g(x)=x2
j(x)=(x + 3)2 is 3 units to the left of g(x)=x2

6. Transformations on Quadratic Graphs
Dilate = to change size / stretch
Multiplying by a number > 1 creates a vertical stretch
Makes the parabola more narrow
Changes the a value in standard form
c(x)=2x2 is more narrow than g(x)=x2 (vertically stretched)
Multiplying by a number between 0 and 1 (a fraction less than 1) creates a horizontal stretch
Makes the parabola wider
c(x)= 𝟏 𝟐 x2 is wider than g(x)=x2 (horizontally stretched)
Multiplying by a negative flips the parabola to open down (changes sign of a value)

7. Patterns in the tables of values…
Table A
Table B
Do Tables A and B …
have the same y-intercepts?
have the same x-intercepts?
Have the same maximum / minimum?
Both open up?
Are both symmetric? x
f(x) -1 8 3 1 2 4 5 x
f(x) -5 -4 -3 3 -2 4 -1 1
What patterns do you see in the tables of values?
How can we know the x-coordinate of the vertex by looking at two symmetric points?
How can you tell if they will open up or down and have the same end behavior without graphing?