[Video: Introduction to Nonlinear Functions]
Learn about quadratic, exponential, and other nonlinear functions that appear frequently on the SAT, and how to interpret their key features.
What are Nonlinear Functions?
Functions whose graphs are not straight lines. Their rates of change are not constant.
Common Types
- Quadratic (parabolas)
- Exponential (growth/decay)
- Square root
- Absolute value
- Cubic
Key Differences from Linear
- Variable rate of change
- Curved graphs
- More complex equations
- Different applications
Quadratic Functions
[Graph of a parabola showing vertex, axis of symmetry, and roots]
Figure 1: Key features of a quadratic function
Standard Form
f(x) = ax² + bx + c
Key Features:
- Vertex: The turning point (minimum/maximum)
- Axis of symmetry: x = -b/(2a)
- Roots: Solutions to ax² + bx + c = 0
- Direction: Opens up if a > 0, down if a < 0
Exponential Functions
[Graphs of exponential growth and decay]
Figure 2: Exponential growth vs. decay
Standard Form
f(x) = a·b^x
Key Features:
- Growth: When b > 1
- Decay: When 0 < b < 1
- y-intercept: (0, a)
- Asymptote: y = 0 (usually)
Other Nonlinear Functions
Square Root Functions
f(x) = a√(x-h) + k
- Domain: x ≥ h
- Starts at (h, k)
- Curves gradually
Absolute Value Functions
f(x) = a|x-h| + k
- V-shaped graph
- Vertex at (h, k)
- Sharp turn at vertex
Key Takeaways
- Recognize nonlinear functions by their equations and graphs
- For quadratics, identify vertex and direction of opening
- For exponentials, determine growth/decay and y-intercept
- Understand domain restrictions (like square roots)
- Practice converting between different forms
Practice Question
The function f is defined by f(x) = 2^x + 1. Which of the following statements is true about the graph of y = f(x) in the xy-plane?
A) The graph is a parabola with vertex at (0,1)
B) The graph has a horizontal asymptote at y = 1
C) The graph decreases as x increases
D) The graph has a vertical asymptote at x = 0
Consider these points:
- What type of function is f(x) = 2^x + 1?
- What are the characteristics of this function family?
- How does the "+1" affect the basic exponential graph?
- Which options describe features of exponential functions?
Analysis:
1. f(x) = 2^x + 1 is an exponential function (base > 0), not a parabola
2. The basic exponential 2^x has a horizontal asymptote at y=0. The "+1" shifts this up to y=1
3. Since the base (2) is greater than 1, the function increases as x increases
4. Exponential functions don't have vertical asymptotes
The correct answer is B) The graph has a horizontal asymptote at y = 1.