Nonlinear functions change at a nonconstant rate. On the SAT, their equations and graphs reveal features such as zeros, vertices, growth factors, intercepts, and intervals of increase or decrease.
Essential concepts
\[f(x)=a(x-h)^2+k qquad g(x)=ab^x\]
The quadratic vertex is (h, k). For an exponential function, a is the value at x = 0 and b is the factor for each one-unit increase in x.
SAT strategy and application
Read a quadratic from vertex form
For \(f(x)=(x-2)^2+1\), identify the vertex and minimum value.
- Match vertex form
\(h=2\) and \(k=1\).
- Use the sign of a
The coefficient of the squared term is positive, so the parabola opens upward.
Vertex: \((2,1)\); minimum value: \(1\).
Mistakes and traps
Check your understanding
For \(g(x)=5(1.2)^x\), what is the initial value?
- 1.2
- 5
- 6
- 20%
Show answer and explanation
Answer: \(5\)
At \(x=0\), \(g(0)=5(1.2)^0=5\).
Key takeaways
What to remember
- Use equation form to reveal the feature the question requests.
- Quadratics have constant second differences; exponentials have constant ratios.
- Interpret parameters through graph features and context.
- Distinguish additive rates from multiplicative growth factors.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.