Transformations of Exponential Functions

Analyze and graph exponential growth and decay functions. Grade 12 Math 30-1.

Lesson 3.2 of Exponents & Logarithms in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Transformation Equation. Learn the complete transformation equation f(x) = a(c)^(b(x-h)) + k. Understand what each parameter does, why c is not a transformation parameter, and how transformations affect domain, range, and asymptote.
• Understanding Each Parameter in Depth. Explore each transformation parameter with visual examples. Learn how k affects the asymptote, how a stretches and reflects, how h shifts horizontally, and how b compresses the graph. Includes factoring techniques and a complete summary table.
• Detailed Examples — Identifying and Applying Transformations. Work through a complete example of identifying all parameters from an equation. Learn how to read a, b, h, and k values, determine domain, range, and asymptote, and understand which parameters affect each feature.
• Sketching Transformed Exponential Graphs. Master the mapping notation technique to sketch transformed graphs without a calculator. Learn to map key reference points, factor exponents correctly, and find equations from graphs using asymptotes and intercepts.

Inside the lesson — a free preview

Building on What You Know

In a previous unit you studied transformations of functions — how parameters a, b, h, and k can stretch, reflect, and shift any graph. Those same rules apply here. The only difference is that the function we are transforming is now an exponential function.

When we apply all four parameters to a base exponential function f(x) = c^x, we get the fully transformed equation:

f(x) = a(c)^(b(x-h)) + k

This single equation contains everything you need to describe any transformation of an exponential function. Let's break down exactly what each parameter does before we start using it.

What Each Parameter Does

a — Vertical Stretch or Reflection. The parameter a stretches the graph vertically. If a = 3, every y-value is multiplied by 3, making the graph three times taller. If a is negative, the graph is also reflected over the x-axis — it flips upside down.

b — Horizontal Stretch or Reflection. The parameter b stretches the graph horizontally. If b = 3, the graph is compressed horizontally by a factor of (1)/(3). If b is negative, the graph is reflected over the y-axis. This parameter is trickier than the others and will be explored in detail later.

h — Horizontal Translation. The parameter h shifts the graph left or right. If h = 2, the graph moves 2 units to the right. If h = -3, the graph moves 3 units to the left. Be careful — the formula says (x - h), so the direction is opposite to the sign you see.

k — Vertical Translation. The parameter k shifts the graph up or down. If k = 4, the graph moves 4 units up. If k = -3, the graph moves 3 units down. This is also the value of the horizontal asymptote after the transformation.

Worked example: Detailed Example 1: Reading All Parameters from an Equation

Consider the function g(x) = 3^(x+2) - 3.

a) What is the original base function f(x)?

b) State the values of a, b, h, and k, and describe the transformation each one represents.

c) State the domain, range, and equation of the horizontal asymptote.

d) Which parameters would affect the domain, range, and horizontal asymptote?

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