Practice Function Families in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

y = mx + b is a family of lines. Different m and b give different lines.

Example 1

easy
The family of quadratics f(x)=ax^2 (with a\neq0) all share the same vertex at the origin. Describe how changing a from 1 to 4 to -2 affects the graph.

Example 2

medium
The family f_k(x)=\dfrac{1}{x-k} has a vertical asymptote at x=k for each parameter k. Analyze the graphs for k=-2, 0, 3 and describe the pattern.

Example 3

easy
In the family y=b^x (exponential), how does the graph change as b varies? Compare b=2, b=1, b=0.5 at x=2.

Example 4

hard
The family y=x^n for integer n\geq1 has different behavior depending on whether n is even or odd. Classify and explain the symmetry of y=x^2, y=x^3, y=x^4, y=x^5.
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