Graphs of functions

Maths · Algebra and graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

To draw the graph of a function, make a table of values, plot the points and join them with a smooth curve. You should know the shapes: a quadratic is a U or an upside-down U, a cubic has an S shape, a reciprocal graph such as y = 6/x has two separate branches, and an exponential graph such as y = 2^x rises faster and faster.

Graphs solve equations. The solutions of f(x) = 0 are the x-values where the curve crosses the x-axis. To solve a different equation, draw a suitable straight line and read the x-values where it meets the curve.

The gradient of a curve at a point is the gradient of the tangent drawn there.

Rules to remember

Roots of f(x) = 0 are the x-coordinates where y = f(x) crosses the x-axis
To solve f(x) = k, draw the horizontal line y = k and read the x-coordinates of the intersections
To solve another equation, rearrange it so the drawn function is on one side; the other side is the line to draw
y = a/x has two branches and never touches the axes: there is no value at x = 0
y = b^x passes through (0, 1); it shows growth when b > 1 and decay when 0 < b < 1
For y = a × b^x the starting value (x = 0) is a, and y is multiplied by b each time x goes up by 1
gradient of a tangent = change in y ÷ change in x, using two points far apart on the tangent

Common mistake

In a table of values, a negative x must be squared with brackets: (−2)² = +4, not −4. Also join points with a smooth curve, never with a ruler.

Worked example

The graph of y = x³ − 2x is drawn. Which straight line must be drawn to solve x³ − 3x − 1 = 0?

The left side must become x³ − 2x, so add x + 1 to both sides.
Left: x³ − 3x − 1 + x + 1 = x³ − 2x. Right: 0 + x + 1 = x + 1.
So x³ − 2x = x + 1. Draw y = x + 1 and read the x-coordinates where it meets the curve.

Answer: The line y = x + 1

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