Common function graphs

There are some common function graph shapes that you should know about.

This page lists some of them.

Common graph terminology

• The y-intercept is the point where the graph crosses the y-axis (where x=0).
• The x-intercepts (or roots) are the points where the graph crosses the x-axis (where y=0).
• Turning points are points where the graph changes from increasing (positive gradient) to decreasing (negative gradient), or the other way round. It is also sometimes called the vertex. The gradient of the line at a turning point is 0.
• An asymptote is a line that the graph approaches as x gets infinitely big or small, but never touches.

Quadratics

• Quadratics are the shape of a parabola.
• They have the general form of y=ax^2 + bx + c.
• The line that goes through the vertex and divides the graph into two symmetrical halves is called the axis of symmetry.
• This is because parabolas are symmetrical (with the line of symmetry going through the vertex).
• The y-intercept is at the point (0, c).

Cubics

• Cubics have the general form of y=ax^3 + bx^2 + cx + d.
• They can have up to 2 turning points.
• The y-intercept is at the point (0, d).
• Cubics can have 1, 2, or 3 x-intercepts (roots).

Quartics

• Quartics have the general form of y=ax^4 + bx^3 + cx^2 + dx + e.
• They can have up to 3 turning points.
• The y-intercept is at the point (0, e).
• They can have 1, 2, 3, or 4 x-intercepts (roots).

Exponentials

• Exponential functions have the general form of y=ab^x, where b>0 and b \neq 1.
• They have a horizontal asymptote at y=0 (the x-axis).
• The y-intercept is at the point (0, a).

Reciprocals

• Reciprocal functions have the general form of y=\frac{a}{x}.
• They have two asymptotes: a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis).
• The graph is in two separate parts (called branches), one in the first and third quadrants if a>0, and one in the second and fourth quadrants if a<0.
• There is no y-intercept or x-intercept, since the graph never touches either axis.