Exploring Quadratic graphs

by

Julie Anne Laycock

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Fix two values of **a, b, c**. Make 5 graphs on the same axes as you vary the third value.

First, let's start by looking at what happens when we set **a** and **b** but change the third value **c**.

I chose to fix **a**=1 and **b**=1 while changing **c**=1,-1,3,-3,5,-5. As we can see changing** c** shifts the graph up and down. When **c** is positive the graph shifts up and when **c** is negative the graph shifts down.

The animated graph shows the graph shifting up and down as **c** ranges from -10 to 10.

Now let's look at what happens when we set **a** and **c** but change the second value **b**.

I chose to fix **a**=1 and **c**=1 while changing **b**=1,-1,3,-3,5,-5. As we can see changing **b** shifts the graph left and right as well as up and down.

The animated graph on the right shows the path of the parabola as **b** ranges from -10 to 10. We can see that the graph appears to follow a path that appears to be a parabola. We also observe that the graph is always crossing the y-axis at (0,1). When x is zero the y-intercept will always be **c**. This is why the parabola is always sliding through this point.

Finally, let's look at what happens when we set **b** and **c** but change the first value **a**.

I chose to fix **b**=1 and **c**=1 while changing **a**=1,-1,3,-3,5,-5. As we can see changing **a** stretches and flattens the graph. When **a** is a number greater than one the graph stretches more and more as the number gets bigger. When **a** is a number less than one the graph also stretches but it also flips.

We can in the animated graph as **a** moves closer and closer to zero and falls between one and negative one the parabola flattens until it reaches zero where the parabola becomes a straight line.