Linear vs. nonlinear functions Knowledge-Based Download PDF

Questions

1. 1.

   Which of the following statements is true regarding the secant line passing through any two distinct points on the graph of a linear function?

   • A) The slope of the secant line changes as the horizontal distance between the two points increases.
   • B) The secant line intersects the graph of the function at exactly one additional point.
   • C) The secant line is perpendicular to the horizontal axis regardless of which points are chosen.
   • D) The secant line completely coincides with the graph of the function.
2. 2.

   The average rate of change of a function $f$ is $4$ over the interval $[1, 2]$ and $10$ over the interval $[2, 3]$. Which of the following conclusions must be true regarding $f$?

   • A) $f$ is a linear function with a slope that increases between intervals.
   • B) $f$ is a linear function because both calculated rates of change are positive.
   • C) $f$ is a nonlinear function because its average rate of change is not constant.
   • D) $f$ is an exponential function because the rate of change increases by a constant difference.
3. 3.

   Which of the following conditions guarantees that a real-valued function $f$ is linear?

   • A) $f(x)$ has both an $x$-intercept and a $y$-intercept.
   • B) $f(x)$ increases at an increasing rate as $x$ increases.
   • C) $f(x)$ changes by a constant multiplicative factor over equal intervals of $x$.
   • D) The average rate of change of $f(x)$ is constant across all closed intervals $[a, b]$ where $a \neq b$.
4. 4.

   Which of the following sets of ordered pairs represents a nonlinear function?

   • A) $\{(1, 1), (2, 4), (3, 9), (4, 16)\}$
   • B) $\{(1, 2), (2, 5), (3, 8), (4, 11)\}$
   • C) $\{(1, 7), (2, 4), (3, 1), (4, -2)\}$
   • D) $\{(1, -4), (2, -2), (3, 0), (4, 2)\}$
5. 5.

   Which of the following equations represents a linear function of $x$?

   • A) $f(x) = \frac{3x - 5}{4}$
   • B) $g(x) = \frac{4}{x} - 5$
   • C) $h(x) = 3x^2 - 5$
   • D) $k(x) = 3\sqrt{x} - 5$