Graphs of Quadratic Functions Vertex Form

Understand how altering the values of $a$, $b$ and $c$ in $y = a(x + b)^2 + c$ affects the corresponding graph.

You should be able to work out what each letter does by breaking down the expression into various graph transformations [we tend to assume $a \neq 0$]:

Here is one way of doing this [when $a > 0$]:

And here is another subtly different way

In the first we used $f(\sqrt{a} x)$ [horizontal squash when $a > 0$] and in the second we used $af(x)$ [vertical stretch] Check that you get the same graph in both cases by trying some simple numbers and using a graph sketching package for $a$, eg $a = 4$.

As an aside: note that sometimes we can only use one method, e.g. when $a = -4$ we cannot use $\sqrt{-4}$. But we could introduce an extra step:

But this is a bit cumbersome.

Exercise

Pick some sets of values for $a, b$ and $c$ and then follow though the graph transformations above step by step using a graph sketching package.

The last three sections of the specification have been covered in earlier sections:

• Use differentiation to help determine the shape of the graph of a given function; including finding stationary points (excluding inflexions); and when the function is increasing or decreasing.
• Use algebraic techniques to determine where the graph of a function intersects the coordinate axes; appreciate the possible numbers of real roots a general polynomial can possess.
• Geometric interpretation of algebraic solutions of equations; relationship between the intersections of two graphs and the solutions of the corresponding simultaneous equations.