The table below is for the function $y = \dfrac{x^{3}}{4} - 2x + 1$.

1. Evaluate the function at the missing table entry $x=2$.

For $x=2$,
$y=\frac{2^3}{4}-2(2)+1$
$=\frac{8}{4}-4+1$
$=2-4+1=-1$

1. Use the completed table to identify all coordinates to be plotted.

From the completed table,

$(-3,0.25),\ (-2,3),\ (-1,2.75),\ (0,1),\ (1,-0.75),\ (2,0),\ (3,1.75).$

2. Plot the points accurately on the given grid.

Plot each point on the Cartesian grid at its corresponding $x$-value and $y$-value. For example, plot $(2,0)$ on the $x$-axis and $(1,-0.75)$ three-quarters of a unit below the $x$-axis.

3. Join the plotted points with an appropriate smooth curve over the required interval.

Join the plotted points with a smooth curve from $x=-3$ to $x=3$, not with straight line segments. The curve should pass close to all the plotted points and stop at $x=-3$ and $x=3$.

1. Relate the equation to intersections of the graph with a horizontal line.

The equation $\dfrac{x^3}{4}-2x+1=k$ is solved graphically by drawing the horizontal line $y=k$ and finding where it meets the curve $y=\dfrac{x^3}{4}-2x+1$. For some values of $k$, this horizontal line cuts the curve at only one point, so there is only one solution.

1. Express the given cubic equation in terms of the graphed function.

The graph already drawn is
$y=\dfrac{x^3}{4}-2x+1.$
From
$x^3-9x-4=0,$
divide throughout by $4$:
$\dfrac{x^3}{4}-\dfrac{9x}{4}-1=0.$

2. Rearrange to obtain the equation of the straight line.

Rearrange so that $\dfrac{x^3}{4}-2x+1$ is on one side:
$\dfrac{x^3}{4}-\dfrac{9x}{4}-1=0$
$\dfrac{x^3}{4}-2x+1=\dfrac{x}{4}+2.$
Since the curve is $y=\dfrac{x^3}{4}-2x+1$, the suitable straight line is
$y=\dfrac{x}{4}+2.$

1. Draw the straight line obtained in part (d)(i) on the same graph.

From part (d)(i), draw the line

$y=\frac{x}{4}+2.$

For example, it passes through convenient points such as $(0,2)$ and $(2,2.5)$.

2. Read the $x$-coordinates where the line intersects the cubic curve.

The solutions of

$x^3-9x-4=0$

are given by the intersection points of the curve

$y=\frac{x^3}{4}-2x+1$

and the line

$y=\frac{x}{4}+2.$

From the graph, the two visible intersections occur at approximately

$x\approx -2.7 \quad \text{and} \quad x\approx -0.5.$