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Exponential Graphs

HigherHigher tier onlyAQAEdexcelOCR

Practise exponential graphs with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through recognising exponential graphs, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Exponential graphs get steeper rapidly and never cross the x-axis.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An exponential graph comes from an equation of the form \(y = k^x\), where the variable is in the index rather than the base. This produces growth that accelerates dramatically.

When \(k\) is greater than \(1\), the curve rises slowly at first and then increasingly steeply, which models population growth and compound interest. When \(k\) is between \(0\) and \(1\), the curve falls, modelling decay.

Every exponential graph passes through \((0, 1)\), because any number to the power zero is \(1\). The curve approaches the horizontal axis but never touches it, since a positive base raised to any power stays positive.

Revision notes

Growth and decay

For \(k > 1\) the curve increases; for \(0 < k < 1\) it decreases.

So \(y = 2^x\) grows, doubling each time \(x\) increases by one, while \(y = 0.5^x\) halves and therefore decays.

Key features

Every curve of this form passes through \((0, 1)\), and the horizontal axis is an asymptote.

The curve stays above the axis for all \(x\), because a positive base to any power is positive. Negative \(x\) values give small positive results, not negative ones.

Comparing with quadratics

An exponential eventually grows far faster than any quadratic, even if it starts smaller.

At \(x = 2\), \(x^2 = 4\) and \(2^x = 4\), but by \(x = 10\) the quadratic gives \(100\) while the exponential gives \(1024\).

Key points

  • An exponential has the variable in the index.
  • \(k > 1\) gives growth; \(0 < k < 1\) gives decay.
  • Every curve passes through \((0, 1)\).
  • The horizontal axis is an asymptote.
  • The curve never goes below the axis.
  • Exponential growth eventually outpaces any quadratic.

Worked examples

Example 1

Find \(y\) when \(x = 3\) for \(y = 2^x\).

Working

\[2^3 = 2 \times 2 \times 2\]substitute x = 3
\[= 8\]evaluate the power

Example 2

Find \(y\) when \(x = -2\) for \(y = 3^x\).

Working

\[3^{-2} = \frac{1}{3^2}\]a negative index means the reciprocal
\[= \frac{1}{9}\]the result is small but still positive

Example 3

State the coordinates where \(y = 5^x\) crosses the y-axis.

Working

\[x = 0 \Rightarrow 5^0 = 1\]any number to the power zero is 1
\[(0, 1)\]state the coordinates

Common mistakes

  • Thinking a negative x gives a negative y.

    3⁻² is 1/9, a small positive number. The curve never dips below the axis.

  • Drawing the curve crossing the x-axis.

    The axis is an asymptote, so the curve approaches but never reaches it.

  • Confusing \(2^x\) with \(x^2\).

    In an exponential the variable is the index, not the base.

  • Forgetting the curve passes through (0, 1).

    This is true for every graph of the form y = kˣ.

Exam tips

  • Check whether the base is above or below 1 to decide growth or decay.
  • Remember every such curve passes through (0, 1).
  • Use negative x values to show the curve flattening towards the axis.
  • Draw the curve smoothly and never touching the horizontal axis.

Key terms

Exponential
A function with the variable in the index.
Growth
Increase that accelerates over time.
Decay
Decrease towards zero over time.
Asymptote
A line the curve approaches but never meets.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.