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# Exponential Graphs
- URL: https://www.esheets.io/exponential-graphs/
- Published: 2026-10-17T19:02:41.000Z
- Updated: 2026-10-17T19:02:41.000Z
- Author: Richard Linnington
- Tags: Algebra, Maths

Practise plotting and interpreting exponential graphs at GCSE Higher level. You will work with growth and decay, exact powers, asymptotes and equations of the form *y* \= *abx*.

The questions build from key graph features and tables of values to finding an exponential model from two points. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This eight-question GCSE Higher worksheet develops a secure understanding of exponential growth and decay graphs.

You will evaluate powers, plot points directly on Cartesian grids and determine exact exponential models.

#### What you’ll practise

- Using intercepts, asymptotes and tables of values.
- Plotting smooth growth and decay curves.
- Solving exact powers and bracketing non-exact solutions.
- Finding *a* and *b* in *y* \= *ab* *x*.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

# Exponential Graphs

Answer all eight questions. Plot graph points directly on each grid.

## Topic guide

### What is an exponential function?

In an exponential function, the variable is in the exponent. The basic form is *y* \= *k* *x*, where *k* is the base. A more general form is *y* \= *ab* *x*.

- If *k* or *b* is greater than 1, the graph shows exponential growth.
- If 0 < *k* < 1, or 0 < *b* < 1, the graph shows exponential decay.
- *y* \= *k* *x* crosses the y-axis at (0, 1), because *k*0 \= 1.
- *y* \= *ab* *x* crosses the y-axis at (0, *a*).
- For the positive-base, unshifted functions here, *y* \= 0 is a horizontal asymptote. The curve approaches the x-axis but does not reach it.

### Evaluate and plot

Use the usual power rules. For example, for *y* \= 3*x*, the values at *x* \= −2, −1, 0, 1 and 2 are 1/9, 1/3, 1, 3 and 9\. A negative exponent gives a reciprocal: 3−2 \= 1/32 \= 1/9.

Complete a table, plot each coordinate carefully, then draw one smooth curve through all the points. Do not join neighbouring points with straight line segments.

### Read and solve

Familiar powers give exact solutions. If 4*x* \= 64, then *x* \= 3\. If a target is not an exact power, compare neighbouring powers. Since 32 \= 9 and 33 \= 27, the solution of 3*x* \= 20 lies between 2 and 3.

### Find an exponential model

If *y* \= *ab* *x* passes through (0, 4) and (2, 36), the first point gives *a* \= 4\. Then 36 = 4*b*2, so *b*2 \= 9 and the positive base is *b* \= 3.

When neither point has *x* \= 0, substitute both points and divide the equations. For points (1, 10) and (3, 90): 10 = *ab* and 90 = *ab*3. Dividing gives 9 = *b*2, so *b* \= 3 and then *a* \= 10/3.

### Common mistakes

- Do not treat *k* *x* as *kx*, or put the point (0, 0) on the curve.
- Do not let the curve touch or cross its horizontal asymptote.
- Growth and decay use multiplicative ratios, not additive differences.
- For negative *x*, a larger positive base produces a smaller value.
- In *y* \= *ab* *x*, *a* is the y-intercept and *b* is the base.
- When both x-values are non-zero, divide the two equations to eliminate *a*.

Logarithms can solve more general exponential equations later, but they are not needed in this worksheet.