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 k⁰ = 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 = 4b², so b² = 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 = ab³. Dividing gives 9 = b², 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.