Sometimes tested MM5.1

Exponential Functions and Graphs

This topic covers the fundamental shape and properties of exponential functions, y = a^x. Understanding these graphs is crucial for modelling rapid growth or decay and forms the visual basis for logarithms.

Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • All graphs of the form y = a^x (for a > 0, a ≠ 1) pass through the y-axis at the point (0, 1), because any positive number raised to the power of 0 is 1.
  • When the base a > 1, the graph shows exponential growth. It starts close to the x-axis for negative x and increases rapidly as x becomes positive.
  • When the base is between 0 and 1 (0 < a < 1), the graph shows exponential decay. It starts high for negative x and decreases rapidly towards the x-axis as x becomes positive.
  • The x-axis is a horizontal asymptote for the graph. The curve gets infinitely close to it but never touches or crosses it.
    the line y = 0
  • For a > 1, a larger base 'a' results in a steeper graph for x > 0. For example, y = 5^x rises faster than y = 2^x after they cross at (0, 1).
  • The function y = a^x is only defined for a positive base 'a' to ensure a continuous real-valued curve.

Diagram

GraphGraph with axes x and y = a^x. (0,1)xy = a^x
Exponential functions with base a > 1 show rapid growth (curve passes through (0,1) and increases steeply), while 0 < a < 1 show decay (curve decreases towards the x-axis asymptote). The y-intercept (0,1) is the same for all exponential curves.

Formulae

y = a^x

To describe any relationship where a quantity changes by a constant multiplicative factor for each unit increase in the independent variable.

Definitions

Exponential Function
A function in the form y = a^x, where the base 'a' is a positive constant and the independent variable 'x' is the exponent.
Base (of an exponential)
The constant number 'a' that is raised to a power in the function y = a^x.
Asymptote
A line that a curve approaches but never touches as it tends towards infinity. For y = a^x, the x-axis is a horizontal asymptote.

Worked example

The graphs of y = 4^x and y = (1/2)^x are drawn on the same set of axes. At what value of x do the two graphs intersect?

  1. 1

    To find the point of intersection, we must set the two expressions for y equal to each other:

    4^x = (1/2)^x
  2. 2

    To solve this equation, we need to express both sides with the same base.

    We can express 4 as 22 and 1/2 as 2-1.

  3. 3

    Substitute these into the equation:

    (22)^x = (2-1)^x
  4. 4

    Apply the power rule of indices, (a^m)n = a^(mn):

    2^(2x) = 2^(-x)
  5. 5

    Now that the bases are equal, we can equate the exponents:

    2x = -x
  6. 6

    Solve this linear equation for x:

    3x = 0, which gives x = 0

Answer: x = 0

Common mistakes

  • ×Confusing the shapes for growth (a > 1) and decay (0 < a < 1). A decay curve like y = (1/3)^x is a reflection of the growth curve y = 3^x in the y-axis, because (1/3)^x = 3^(-x).
  • ×Incorrectly stating that the graph passes through the origin (0, 0). All y = a^x graphs pass through (0, 1).
  • ×Assuming the graph can have negative y-values. For any positive base 'a', a^x is always positive, so the graph is always above the x-axis.

No-calculator tips

  • ✓To quickly compare two growth graphs, like y = 2^x and y = 5^x, test a simple positive integer like x=1. Since 51 > 21, the graph of y=5^x is steeper for x>0.
  • ✓Recognise that a fractional base is a negative power of its reciprocal. For instance, (1/5)^x is identical to 5^-x. This helps in visualising transformations.
  • ✓When solving equations, always look for a common base for all terms. Common bases in ESAT are likely to be simple integers like 2, 3, or 5.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 2 topics