Can we plot this graph and use it to solve two equations?

Can we plot this graph and use it to solve two equations?

Question

Using a scale of $2\,\text{cm}$ to one unit on each axis, draw the graph of $y=6x-x^2$ for values of $x$ from $0$ to $6$ inclusive.

Use your graph to solve the equation $6x-x^2=7.$

By adding a suitable straight line to your graph, solve the equation $6x-x^2=8-\tfrac{1}{2}x.$

Solution

Let’s first draw our axes, allowing $2\,\text{cm}$ per unit. We will plot the values of $y$ for $x=0$, $1$, $2$, $3$, $4$, $5$, $6$.

Now let’s join up the points as smoothly as we can.

We’ll add the line $y=7$ to our plot and see where it intersects the curve.

We read off the approximate solutions. From our plot, we see that $x\approx 1.6$ and $x \approx4.4$ solve $6x-x^2=7$.

Using the quadratic formula to solve this equation gives $x\approx 1.586$ and $x \approx4.414$.

Let’s now add the line $y=8-\frac12 x$ to our sketch, making sure to plot at least $3$ points so that we draw it accurately.

We can see from our plot that the smaller solution of $6x-x^2=8-\frac12x$ is slightly larger than the smaller solution of $6x-x^2=7$, and the larger solution is larger than the larger solution of $6x-x^2=7$.

Our graph tells us (your grid will help you) that the solutions of $6x-x^2=8-\frac12x$ are $x\approx 1.6$ and $x\approx 4.9$.

Using the quadratic formula to solve this equation gives $x\approx 1.649$ and $x \approx4.851$.