4.1 Exponential Functions
Ideas in this lesson
- Define and evaluate real exponentials
- Build, compare and identify exponential graphs
- Transform exponential graphs and explain their features
- Compare exponential and power growth in multiple windows
- Distinguish parameter effects and prove difference quotients
- Analyze nonlinear exponential graphs
- Model doubling, cumulative pay and physical scales
- Derive and compare periodic compound interest
- Discount future value and calculate effective yield
- Solve graphical exponential and logarithmic inequalities
Define and evaluate real exponentials · Worked example
For \(f(x)=\left(5\right)^{\dfrac{x}{2}}\), name the displayed base and evaluate at \(-2\), \(0\), \(2\), \(4\), \(\dfrac{1}{2}\), \(\dfrac{3}{10}\), \(\sqrt{3}\), \(\pi\) to three decimals. Explain what distinguishes an exponential from a power function and how irrational powers are interpreted. Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(0.200\), \(1.000\), \(5.000\), \(25.000\), \(1.495\), \(1.273\), \(4.030\), \(12.530\).
\(f(x)=\left(5\right)^{\dfrac{x}{2}}\). The displayed constant base is \(5\). Equivalently, in the form \(C a^x\), the effective base is \(a=\sqrt{5}\). Substitute the entire input into the exponent before applying the outside multiplier.
\(f(-2)=\dfrac{1}{5}\) (exact).
\(f(0)=1\) (exact).
\(f(2)=5\) (exact).
\(f(4)=25\) (exact).
\(f(\dfrac{1}{2})=\sqrt[4]{5}\), approximately \(1.495\).
\(f(\dfrac{3}{10})=5^{\dfrac{3}{20}}\), approximately \(1.273\).
\(f(\sqrt{3})=5^{\dfrac{\sqrt{3}}{2}}\), approximately \(4.030\).
\(f(\pi)=5^{\dfrac{\pi}{2}}\), approximately \(12.530\).
An exponential has a fixed positive base other than 1 and a variable exponent. A positive base defines every real input; base 1 gives a constant, while nonpositive bases do not define this real exponential family.
For an irrational exponent \(\sqrt{3}\), rational inputs \(1.7,1.73,1.732\) give successively better approximations, not an exact finite decimal.
Written-work review:
\(f(-2)=\dfrac{1}{5}\) (exact).
\(f(0)=1\) (exact).
\(f(2)=5\) (exact).
\(f(4)=25\) (exact).
\(f(\dfrac{1}{2})=\sqrt[4]{5}\), approximately \(1.495\).
\(f(\dfrac{3}{10})=5^{\dfrac{3}{20}}\), approximately \(1.273\).
\(f(\sqrt{3})=5^{\dfrac{\sqrt{3}}{2}}\), approximately \(4.030\).
\(f(\pi)=5^{\dfrac{\pi}{2}}\), approximately \(12.530\).
An exponential has a fixed positive base other than 1 and a variable exponent. A positive base defines every real input; base 1 gives a constant, while nonpositive bases do not define this real exponential family.
For an irrational exponent \(\sqrt{3}\), rational inputs \(1.7,1.73,1.732\) give successively better approximations, not an exact finite decimal.
Written-work review:
- Identify the displayed positive base and distinguish the effective base from a variable-base power function.
- Substitute every listed input into the whole exponent and round only final outputs to three decimals.
- Explain irrational powers using rational approximations rather than asserting an exact finite input decimal.