Transformations of Quadratic Functions - KEY
So far we have only discussed problems within the context of gravity. Each of those questions have
had the coefficient of -16 for t2. But does it always have to be -16?
Can you think of an example for a quadratic function in real life that isn’t pulled by gravity?
In this study, we will investigate all quadratic equations. As you go through this study, ask yourself,
How can you predict the shape and location of graphs of quadratic functions with rules in the form
y = ax2 + c?
In your graphing calculator look at each set of equations by placing them into y 1, y2, y3…
See if you can determine how we each graph is going to move and why.
Set 1
Set 2
y = x2
y = -x2
y = x2 + 3
y = -x2 + 5
y = x2 – 4
y = -x2 - 1
1. Compare the graphs of y = x2 and y = x2 + 3. How are they similar? How are they different?
They are the same shape, but x2 + 3 moves up 3 units on the y-axis.
2. Compare the graphs of y = x2 and y = -x2 - 1. How are they similar? How are they different?
The graphs of x2 and -x2 – 1 are the same shape, but -x2 – 1 is shifted down and in the
opposite direction.
3. Explain how your answers in questions 1 and 2 help to explain the patterns relating the types
of quadratics you see in each set?
Adding or subtracting a value to x2 makes the graph shift up or down, and making x2
negative makes it reflect over the x-axis.
4. Look at graphs of functions given for Set 1 to see if you notice any patterns that relate the
values of a and c in the rule y = ax2 + c to help find the location of the x intercepts, y intercepts
and maximum or minimum point. Write down your thoughts here:
The y –intercept is the value that is added or subtracted, or c.
The minimum point for each graph here is (0,0).
None of these graphs have x-intercepts except x2
5. Do your thoughts match for Set 2?
Look at graphs of functions given below to see if you notice any patterns that relate the values of a
and b in the rule y = ax2 + bx to help find the location of the x intercepts, y intercepts and maximum or
minimum point. It might help to think about each function using the equivalent factored form y =
x(ax +b). Think about the equivalent factored form as the step before you distribute!
The first one has been done for you.
Set 1
y = x2
Set 2
y = x(x)
y = -x2
y = -x(x)
y = x2 + 4x y = x(x + 4)
y = -x2 + 5x y = -x(x + 5)
y = x2 – 4x y = x(x - 4)
y = -x2 – 5x y = -x(x - 5)
a. X intercepts
Help students to find the x-intercept of each equation by setting the two linear factors equal
to 0.
b. Y intercepts
All of the y-intercepts are 0.
c. Maximum or Minimum Point
The maximum or minimum is always the average x-value of the two x-intercepts.
Putting Everything Together
So far, you have learned how to predict the patterns in graphs for three special types of quadratic
functions: y = ax2 , y = ax2 + c, and y = ax2 + bx
Using what you know about the patterns you’ve already found, see if you can apply those same
patterns to the quadratic graphs below.
The diagram at the right gives graphs for three of the four quadratic functions below.
y = x2 – 4x
y = x2 – 4x + 6
y = – x2 – 4x
y = x2 – 4x – 5
Without using your calculator:
Determine the function with the graph that is missing on the diagram.
y = – x2 – 4x
Match each function to their graphs, and explain your reasoning for how they match.
Graph II: y = x2 – 4x
Graph I: y = x2 – 4x + 6
Graph III: y = x2 – 4x – 5
What would the graph of the missing function look like? What characteristics would you expect
to see in the graph?
The graph of y = – x2 – 4x would have x-intercepts at x = 0, and x = -4, and would not shift up
and down any.