# Graph Area

Can you find the area between this graph and the x-axis, between x=3 and x=7?

What is the area between the lines $y=2x-6$, $x=3$, $x=7$ and the $x$-axis?

*This problem is adapted from the World Mathematics Championships*

**Plotting the graph using a table of values**

If we plot the graph first, then we will be able to see the shape whose area we are looking for.

We can use a table of values to plot the graph, and we are interested in $x$ values between $3$ and $7$, so going from $2$ to $8$ will help us see the whole shape.

If $x=2$, then $y=2\times2-6=4-6=-2$. The rest of the values can be found in the same way.

$x$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |

$y$ | $-2$ | $0$ | $2$ | $4$ | $6$ | $8$ | $10$ |

The graph looks like this:

Image

We need to find the area between $x=3$ and $x=7$, which is coloured green below.

Image

The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.

**Sketching the graph using relevant points**

From the equation $y=2x-6$, we know that the graph will be a straight line.

Since we are interested in the area between $x=3$ and $x=7$, we could check where the graph is at those two points.

When $x=3$, $y=2\times3-6=6-6=0$.

When $x=7$, $y=2\times7-6=14-6=8$.

So $(3,0)$ and $(7,8)$ lie on the graph.

Image

So the graph must be the straight line through those points, and the area required is the green area, as shown below.

Image

The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.