Using a scale of \(\quantity{2}{cm}\) to one unit on each axis, draw the graph of \[y=6x-x^2\] for values of \(x\) from \(0\) to \(6\) inclusive.
Let’s first draw our axes, allowing \(\quantity{2}{cm}\) per unit. We will plot the values of \(y\) for \(x=0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\).
![A graph with the points (0,0), (1,5), (2,8), (3,9), (4,8), (5,5), (6,0) marked.](/geometry-of-equations/r7296/images/sketch1.png)
Now let’s join up the points as smoothly as we can.
![A graph with the same points marked, with a smooth curve passing through them all.](/geometry-of-equations/r7296/images/sketch2.png)
Use your graph to solve the equation \[6x-x^2=7.\]
We’ll add the line \(y=7\) to our plot and see where it intersects the curve.
![The same curve drawn, with the horizontal line y = 7 drawn, and vertical lines drawn down to the x axis from the two points where the line and the curve cross.](/geometry-of-equations/r7296/images/sketch3.png)
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 \(x \approx4.414\).
By adding a suitable straight line to your graph, solve the equation \[6x-x^2=8-\tfrac{1}{2}x.\]
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.
![Graph with the original curve, with points of y = 8 - x over 2 marked and the line drawn through them. Again vertical lines are drawn down to the x axis from the two points where the line and the curve cross.](/geometry-of-equations/r7296/images/sketch4.png)
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 \(x \approx4.851\).