Take a look at this diagram.

Graphs of y=f(x), y=10-3f(x) and y=3f(3x-45) drawn on the same axes. The intercepts of y=3f(3x-45) are 15, 25 and 30. The largest intercepts of y=f(x) is also at x=15. A region between y=f(x) and the x axis has been shaded blue. A region between y=10-3f(x) and the x axis has been shaded orange.

The area of the blue region is \(53\) and the area of the orange region is \(125\).

Use this to explain why \[\int_{15}^{30}3f(2x-45)\,dx=67.\]

Here are some things to think about while looking at the diagram.

  • What transformations of \(y=f(x)\) give us the other two graphs?

  • What are the \(x\)-intercepts of \(y=f(x)\)?

  • How long are the vertical edges of the orange region?

  • How could you express the area of the orange region in terms of areas underneath the untransformed blue graph?

You may find it helpful to explore what happens when you move the sliders in this GeoGebra file.

Can you make the purple graph line up with the black graph?

The purple graph is a transformation of \(y=f(x)\) and this transformation involves three parameters, \(a\), \(b\) and \(c\). The parameters are related to scaling and translation, but the numerical value of the parameter \(b\) is not necessarily the same as the value the graph is translated by along the \(x\)-axis.

You may like to explore what the translation is for the same value of \(b\) but different values of \(a.\)

What if we applied these ideas to the orange curve too?

Now try to evaluate each of the following integrals.

  1. \(\int_{-4}^{-2}\sqrt{x+5}\,dx\)

  2. \(\int_{0.2}^{1.3}e^{2x}\,dx\)

Think about what is happening in these GeoGebra files as you move the sliders.

How could this help you to evaluate the integrals?