Food for thought

## Solution

Here are the graphs of four functions. The equations of the graphs are $y=f(x) \quad y=f(x)-8 \quad y=3f(x) \quad \text{and} \quad y=3f(x)+8.$

• The $x$-coordinates of $A$, $B$, $C$ and $D$ are all the same. What can you deduce about the gradients of the curves at $A$, $B$, $C$ and $D$?

• The gradient of the tangent at $D$ is $\tfrac{1}{4}$. What are the gradients of the tangents at $A$, $B$ and $C$?

Here are the graphs of another four functions. The equations of these graphs are $y=f(x)\quad y=f(x+20)-8 \quad y=3f(x-25) \quad \text{and} \quad y=-3f(x)+10.$

• The $x$-coordinates of points $E$, $F$, $G$ and $H$ are $-15$, $5$, $5$ and $30$ respectively. What can we say about these points?

• The gradient of the tangent at $E$ is $-\tfrac{1}{4}.$ What are the gradients of the tangents at $F$, $G$ and $H$?

The transformations of these curves look a bit more complicated than those in the first set of functions, but we can use similar ideas to work out which curve is which and find the coordinates of points $E$, $F$, $G$ and $H$.

We have used the idea that when a graph is translated, the tangents at corresponding points have the same gradient. It is important to note that this is not the same as saying that translations do not affect gradient functions.

Can you think of an example where the translation doesn’t affect the gradient function, and an example where it does?