Plotting a function turns an algebraic rule into a picture, letting you read off roots and intersections that would otherwise need algebra to find exactly.

Substitute each given $x$-value in turn, taking particular care with negative $x$-values inside powers or reciprocal terms, where sign errors are easy to make.

For $y=x^2+\dfrac{2}{x}$:

Functions containing a term like $\dfrac{1}{x}$ or $\dfrac{a}{x^2}$ are undefined at $x=0$, since division by zero has no value. The table of values will usually leave that column blank, and the plotted curve must never be joined across this gap — the two branches either side of $x=0$ are drawn as separate curves that approach, but never touch, the vertical axis.

Recognise that solving $f(x)=k$ means finding the $x$-coordinates where the drawn curve $y=f(x)$ crosses the horizontal line $y=k$. Draw that line on the same axes and read off the intersection points.

To solve $x^2+\dfrac{2}{x}=5$ using the curve above, draw the line $y=5$ and read the $x$-values where it crosses the curve: approximately $x\approx-0.4$ and $x\approx2.2$.

Rearrange the equation so that everything on one side matches the curve exactly as plotted, leaving the simplest possible expression on the other side to draw as an extra line.

To solve $x^2+\dfrac{2}{x}-3x=0$ using the same plotted curve $y=x^2+\dfrac{2}{x}$, rearrange to $x^2+\dfrac{2}{x}=3x$ and draw the line $y=3x$ — not some more complicated rearrangement.

Worked Example

The curve $y=x^3-4x$ has already been drawn for $-3\le x\le3$.

(a) By drawing a suitable straight line on the same grid, find how many solutions the equation $x^3-4x-2=0$ has.

(b) State the equation of the line you would draw.

1. Rearrange $x^3-4x-2=0$ so that the left side matches the plotted curve exactly:

1. The required line is the horizontal line $y=2$.
2. A cubic curve of this shape crosses a horizontal line drawn between its local maximum and minimum values at three separate points, so reading off the intersections gives 3 solutions.