STAAR Algebra I: a complete guide to exponential functions and equations

What this category demands

The Exponential Functions and Equations reporting category (TEKS A.3C, A.9) is about 10 percent of the STAAR Algebra I test, the smallest function category, but it is a reliable discriminator above the Approaches standard and connects to compound interest in the Number and Algebraic Methods category. The skills are: write growth and decay models, graph them and read the asymptote, solve simple exponential equations, distinguish linear from exponential, and apply the model. Each dot-point page has its own practice: exponential growth and decay models, graphing exponential functions, solving exponential equations and linear versus exponential, and exponential applications and best fit.

Writing growth and decay models

An exponential function is $f(x) = ab^x$: $a$ is the initial value (at $x = 0$), and $b$ is the per-step factor. For a percent rate $r$, growth uses $b = 1 + r$ ($b > 1$) and decay uses $b = 1 - r$ ($0 < b < 1$). So 20% growth gives $b = 1.2$ and 15% decay gives $b = 0.85$. The form is not on the reference sheet, so memorize it, and read the base back as a rate ($r = b - 1$ for growth, $1 - b$ for decay) when asked.

Graphing and the asymptote

The graph of $f(x) = ab^x$ has a $y$-intercept at $(0, a)$ and a horizontal asymptote at $y = 0$, which it approaches but never crosses. Growth ($b > 1$) rises steeply; decay ($0 < b < 1$) falls toward the asymptote. The domain is all real numbers and the range is $y > 0$ (for $a > 0$), unlike a parabola's vertex-bounded range.

Solving and applications

Solve a simple exponential equation by rewriting both sides with a common base, then setting exponents equal: $2^{x+1} = 16 = 2^4$ gives $x = 3$. For applications (population, depreciation, compound interest), evaluate the model by raising $b$ to the power $t$, not multiplying by $t$. Compound interest $A = P(1 + r)^t$ is itself an exponential with $a = P$ and $b = 1 + r$.

How this category is examined

• Multiple choice. Interpret $a$ or $b$, classify growth or decay, find a $y$-intercept, or evaluate a model. The "rate versus factor", "growth versus decay", and "linear instead of exponential" distractors are standard.
• Equation editor and number entry. Write a model, solve a common-base equation, or compute a future value.
• Inline choice. Choose growth or decay, the asymptote, or linear versus exponential.

Check your knowledge

Work these as you would for credit on the redesigned test.

1. Write a model for $2,000 growing 4% per year. (1 point)
2. Write a model for a $15,000 car losing 12% of its value per year. (1 point)
3. State the $y$-intercept and asymptote of $f(x) = 8(2)^x$. (1 point)
4. Is $f(x) = 5(0.7)^x$ growth or decay, and what is its range? (1 point)
5. Solve $3^{x-1} = 27$. (1 point)
6. A table shows $2, 6, 18, 54$ for inputs $0, 1, 2, 3$. Linear or exponential? Write a function. (2 points)
7. A culture of 100 doubles every hour. How many after 5 hours? (2 points)
8. A $5,000 investment grows 6% compounded annually. Value after 2 years, to the nearest dollar? (2 points)