The variables x and y are connected by the equation $y = 2x^3 - 21x^2 + 54x$. Some corresponding values of x and y are given in the table below.

1. Evaluate $y$ when $x=4.5$, since $p$ is the corresponding $y$-value in the table.

For $x=4.5$,
$p=2(4.5)^3-21(4.5)^2+54(4.5).$
Now $(4.5)^2=20.25$ and $(4.5)^3=91.125$, so
$p=2(91.125)-21(20.25)+54(4.5)$
$=182.25-425.25+243=0.$

1. List the coordinates to be plotted from the table.

Using $p=0$, the points are
$(0,0),\ (0.5,22),\ (1,35),\ (2,40),\ (3,27),\ (4,8),\ (4.5,0),\ (5,-5),\ (6,0).$

2. Plot all the points accurately on the given grid.

Mark each point at its corresponding $x$-coordinate and $y$-coordinate on the grid. For example, $(0.5,22)$ is halfway between $x=0$ and $x=1$, and 2 units above $y=20$.

3. Join the plotted points with a suitable smooth curve.

Draw one continuous smooth curve through the plotted points, following the shape suggested by the values. The curve rises from $(0,0)$ to near $(2,40)$, then falls through $(4.5,0)$ to $(5,-5)$, and rises again to $(6,0)$.

1. Identify the highest point of the curve on the given graph.

From the smooth curve, the highest point occurs between $x=1$ and $x=2$, slightly above $y=40$.

So the maximum value of $y$ is about
$41.$

2. Identify the lowest point of the curve on the given graph.

From the smooth curve, the lowest point occurs near $x=5$, slightly below the $x$-axis.

So the minimum value of $y$ is about
$-5.$

1. Draw the tangent at the point $(0.5,22)$ on the curve.

On the completed graph, draw a straight line that just touches the curve at $(0.5,22)$ and follows the direction of the curve at that point.

2. Find the gradient of the tangent using vertical change over horizontal change.

Choose two well-spaced points on the tangent and calculate

$\text{gradient}=\frac{\text{vertical change}}{\text{horizontal change}}.$

From an accurately drawn tangent, this gives approximately

$\text{gradient}\approx 35.$

1. Find two points on the line $y=45-6x$ for the given range of $x$.

When $x=0$,
$y=45-6(0)=45$.

When $x=6$,
$y=45-6(6)=9$.

So two points on the line are $(0,45)$ and $(6,9)$.

2. Draw the required straight line on the same axes.

Plot $(0,45)$ and $(6,9)$ on the same graph as the curve, then join them with a straight line segment for $0\le x\le 6$.

1. Locate the intersection points of the line and the curve on the same graph.

From the graph, mark the two points where the straight line $y=45-6x$ meets the curve $y=2x^3-21x^2+54x$.

2. Read off the corresponding $x$-coordinates.

Reading vertically down from the two intersection points to the $x$-axis gives

$x \approx 1.2 \quad \text{and} \quad x \approx 3.0.$