ACT Functions and Graphs Quiz Quiz

Challenge your understanding of ACT functions and graphs concepts with this engaging quiz. Sharpen your skills on interpreting, analyzing, and manipulating mathematical functions and their graphical representations—ideal for ACT math preparation.

1. Identifying Function Type

   Which of the following equations represents a linear function whose graph is a straight line passing through the point (0, 3)?

   1. y = 3/x
   2. y = |x| + 3
   3. y = x^2 + 3
   4. y = 2x + 3

   Explanation: The equation y = 2x + 3 represents a linear function because it is in the form y = mx + b, which graphs as a straight line. y = x^2 + 3 is a quadratic, so it forms a parabola, not a straight line. y = 3/x is a rational function, resulting in a hyperbola. y = |x| + 3 creates a 'V' shaped absolute value graph, not a straight line.

2. Function Transformation

   If the graph of f(x) = x^2 is shifted 4 units up, which equation represents the new function g(x)?

   1. g(x) = (x - 4)^2
   2. g(x) = x^2 - 4
   3. g(x) = x^2 + 4
   4. g(x) = (x + 4)^2

   Explanation: Adding 4 outside the function, as in g(x) = x^2 + 4, results in a vertical shift 4 units upward. (x + 4)^2 and (x - 4)^2 represent horizontal shifts, not vertical. Subtracting 4, as in x^2 - 4, would move the graph 4 units down rather than up.

3. Reading Graph Intercepts

   Given a function whose graph passes through the point (2, 0), which statement is true about the function?

   1. The maximum of the function is at (2, 0).
   2. The function has a y-intercept at y = 2.
   3. The graph is undefined at x = 2.
   4. The function has an x-intercept at x = 2.

   Explanation: If the graph passes through (2, 0), the function crosses the x-axis at x = 2, which is the x-intercept. A y-intercept would require the point (0, y). The maximum is not necessarily at (2, 0) unless specified, and the function is defined at x = 2 since the point exists.

4. Recognizing Function Notation

   If h(x) = 2x - 5, what is the value of h(4)?

   1. 8
   2. -3
   3. 3
   4. 13

   Explanation: Substituting x = 4 gives h(4) = 2(4) - 5 = 8 - 5 = 3. The answer -3 could result from calculating 2 minus 5 (incorrect substitution), while 8 ignores subtracting 5, and 13 comes from incorrectly adding rather than subtracting.

5. Function Domain

   What is the domain of the function f(x) = 1/(x - 3)?

   1. All real numbers
   2. All positive numbers
   3. All real numbers except x = 3
   4. All real numbers except x = 0

   Explanation: The function f(x) = 1/(x - 3) is undefined when x - 3 = 0, which is when x = 3. Excluding only x = 3 gives the correct domain. Excluding x = 0 applies to functions like 1/x, not this one. The function is defined for negative values, so 'all real numbers' and 'all positive numbers' are incorrect.