4.1 Exponential functions and transformations

An exponential function places the variable in an exponent. Its constant multiplicative change distinguishes it from a linear function’s constant additive change.

Exponential function. A function \(f(x)=ab^x\), where \(a\ne0\), \(b>0\), and \(b\ne1\).

4.1.1 Basic graphs and properties

For \(f(x)=b^x\), the domain is \(\mathbb R\), the range is \((0,\infty)\), the \(y\)-intercept is \((0,1)\), and \(y=0\) is a horizontal asymptote. The function increases when \(b>1\) and decreases when \(0<b<1\). Equal increases in \(x\) multiply the output by the same factor \(b\).

Worked Example: Identifying an exponential base

An exponential graph passes through \((0,1)\) and \((2,9)\). If \(f(x)=b^x\), then \(b^2=9\). Since an exponential base is positive, \(b=3\) and \(f(x)=3^x\).

Try It. Find \(b\) if \(f(x)=b^x\) passes through \((3,8)\).
Check Your Work \(b^3=8\), so \(b=2\).

Worked Example: Interpreting exponential decay

For \(C(t)=12(0.80)^t\), the initial concentration is \(C(0)=12\). Each time unit multiplies concentration by 0.80, which is a 20% decrease.

Try It. Interpret the initial value and percent change in \(N(t)=450(1.06)^t\).
Check Your Work The initial value is 450 and the quantity grows by 6% per time unit.

4.1.2 Transformations of exponential graphs

The transformation rules developed in the functions chapter apply. In \(y=ab^{k(x-h)}+d\), \(h\) shifts horizontally, \(d\) shifts vertically, \(a\) changes vertical scale and may reflect the graph, and \(k\) changes horizontal scale. The horizontal asymptote moves from \(y=0\) to \(y=d\).

Worked Example: Transforming an exponential graph

For \(f(x)=2(3^{x-1})+2\), shift \(3^x\) right 1, stretch vertically by 2, and shift up 2. The horizontal asymptote is \(y=2\), and \(f(1)=4\).

Try It. Describe \(g(x)=-2^{x+3}+1\).
Check Your Work Shift \(2^x\) left 3, reflect across the \(x\)-axis, and shift up 1. The asymptote is \(y=1\).

Worked Example: Determining a transformed range

For \(y=5-4(0.5)^x\), the exponential term is positive, so \(y<5\). The domain is \(\mathbb R\), the range is \((-\infty,5)\), and the horizontal asymptote is \(y=5\).

Try It. State the range and asymptote of \(y=3(2^x)-7\).
Check Your Work Range \((-7,\infty)\); asymptote \(y=-7\).

Applied Problem: Modelling contaminant decay. A contaminant concentration is initially 18 mg/L and decreases by 15% each day. Write an exponential model, find the concentration after 6 days, and identify the horizontal asymptote.

Check Your Work \(C(t)=18(0.85)^t\). Then \(C(6)\approx6.79\) mg/L. The horizontal asymptote is \(C=0\).

4.1.3 Practice Problems

  1. State the domain and range of \(4^x\).
    Check Your Work Domain \(\mathbb R\); range \((0,\infty)\).
  2. Does \((1/3)^x\) increase or decrease?
    Check Your Work It decreases because \(0<1/3<1\).
  3. Find the horizontal asymptote of \(2(5^x)-6\).
    Check Your Work \(y=-6\).
  4. Find \(f(3)\) for \(f(x)=7(0.5)^x\).
    Check Your Work \(7/8=0.875\).
  5. Write a model with initial value 80 and growth rate 4%.
    Check Your Work \(f(t)=80(1.04)^t\).
  6. Bacterial concentration begins at 240 cells/mL and grows 8% per hour. Write the model.
    Check Your Work \(N(t)=240(1.08)^t\).
  7. Chlorine concentration begins at 2.4 mg/L and retains 92% each hour. Find it after 5 h.
    Check Your Work \(2.4(0.92)^5\approx1.58\) mg/L.
  8. A transformed treatment model is \(R(t)=60-20(0.7)^t\). State its initial value and limiting value.
    Check Your Work \(R(0)=40\) and the limiting value is 60.