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MTH 103 Exam 1

MTH 103 — Applied Precalculus

Exam 1
Spring 2026

Part 1: True-False

The following are true-false questions. Circle true or false accordingly. No justification is required.

  1. If a vertical line crosses the graph of a curve in more than two places, then the curve is a function.
  2. A numerically given function is linear if the slope between any two distinct points is constant.
  3. A quadratic function with a vertex of $(4,5)$ has an axis of symmetry given by $x=4$.
  4. The vertex of a quadratic function is either the maximum value or the minimum value.
  5. The expression $(x+3)^2$ is equal to $x^2+3^2$.
  6. The function $f(x)=3x+4$ is an increasing function.
  7. The function $f(x)=2x+3$ has a horizontal intercept of $(3,0)$.
  8. A function can have two or more vertical intercepts.

Part 2: Multiple Choice / Short Answer

Circle the best answer for each question, or fill in the blank(s).

  1. Given $f(x)=\dfrac{3x-1}{2}$, then $f(x+h)$ is equal to
    • $\dfrac{3x+h-1}{2}$
    • $\dfrac{3x-3h}{2}$
    • $\dfrac{3x+3h-1}{2}$
    • $\dfrac{3x+h}{2}$
  2. Let $C$ denote the number of calories in a candy bar. Let $w$ denote the weight in ounces of the candy bar. Suppose $C$ is a linear function of $w$ so that $C = f(w)$. What are the units for the slope of this function?
    • ounces per calorie
    • calories per ounce
    • calories
    • ounces
  3. Complete this sentence. If the product of three numbers is $0$, then at least one of the numbers is:
  4. Suppose a quadratic function $g(x)$ has horizontal intercepts $(-1,0)$ and $(3,0)$. What can you say, with certainty, about the function?
    • $g(x)$ opens upward
    • $g(x)$ has a vertex at $(-1,0)$
    • $g(x)$ has a line of symmetry of $x=1$
    • $g(x)$ has a vertex of $(3,k)$ for some number $k$
  5. Write a formula for a function $f(x)$ which squares the input, multiplies the result by $3$, then adds $2$.
  6. Four linear functions are given below, labeled “line 1” to “line 4”. Next to each formula write the number ($1$ through $4$) that corresponds to that equation.

    \begin{align*} y=3 & \quad \text{corresponds to which line? } \\[1em] y=x-3 & \quad \text{corresponds to which line? } \\[1em] y=-3x+3 & \quad \text{corresponds to which line? } \\[1em] y=-x & \quad \text{corresponds to which line? } \\[1em] \end{align*}

    Graph showing four lines. Line 1 is decreasing, passing through (1,0) and (2,-3). Line 2 is decreasing, passing through (-3,3) and (-1,1). Line 3 passes through (-3,3) and (2,3). Line 4 passes through (1,-2) and (4,1).
  7. Consider the graph of a function $y=f(t)$ given below.
    Graph of a continuous curve showing f(t) versus t. The curve starts at approximately (0,2), decreases to a minimum at approximately (3,-2), then increases to a maximum at approximately (9,4), and finally decreases to approximately (12,2). The horizontal axis is labeled t with values from 0 to 12, and the vertical axis is labeled f(t).
    The average rate of change of $f(t)$ between $t=0$ and $t=3$ is
    • negative
    • positive
    • zero
    • cannot be determined
    and the average rate of change of $f(t)$ between $t=6$ and $t=9$ is
    • negative
    • positive
    • zero
    • cannot be determined
  8. What is the domain of the function $f(x)=\sqrt{x+2} \,$?
    • All numbers $x$ such that $x\ge -2$.
    • All numbers $x$ such that $x\le -2$.
    • All numbers $x$ such that $x\neq 2$.
    • Only $x=2$.
  9. If the following table describes a linear function, then determine its formula.
    $x$ 1 2 3 4
    $y$ 4.4 4.8 5.2 5.6
    • Not a linear function
    • $y=0.4x+4$
    • $y=2.5x+1.9$
    • $y=0.4x+4.4$
  10. Which of the following describe the linear function that passes through $(1,2)$ and $(5,7)$?
    • $y-2=\dfrac{5}{4}(x+1)$
    • $y-2=\dfrac{5}{4}(x-1)$
    • $y-2=\dfrac{4}{5}(x-1)$
    • $y+2=\dfrac{4}{5}(x-1)$

Part 3: Open Response Problems

Show all your work for each open response problem. No credit will be given for correct answers without supporting work.

  1. The height, measured in feet, of a rock $t$ seconds after it was dropped from a bridge is described by the function $h(t) = -5t^2+80$.
    1. Find the vertical intercept of $h(t)$.
    2. Find the positive horizontal intercept of $h(t)$ (i.e., for $t\ge 0$).
    3. Which of the following best describes the vertical intercept?
      • The height from which the rock was dropped.
      • The amount of time that the rock fell for before hitting the ground.
      • The speed of the rock when it hit the ground.
      • The speed of the rock when it started falling.
    4. Which of the following best describes the positive horizontal intercept?
      • The height from which the rock was dropped.
      • The amount of time that the rock fell for before hitting the ground.
      • The speed of the rock when it hit the ground.
      • The speed of the rock when it started falling.
  2. Factor each of the following quadratics into the form $(x+M)(x+N)$ for numbers $M$ and $N$, or state that they cannot be factored.
    1. $x^2+7x+12$
    2. $x^2-4$
  3. Data is collected for the height $p$ in centimeters (cm) of a plant $t$ months after it was planted. After $4$ months the plant is $110$ cm tall, and after $10$ months the plant is $230$ cm tall. Determine the linear function $p(t)$ that describes the height of the plant after $t$ months.
  4. Rewrite the function $f(x)=(x+3)^2-4$ in standard form.
  5. The graph of a function $y=f(x)$ is given below.
    Graph of a continuous function y=f(x) on a coordinate grid. The curve starts at approximately (0,0), increases to a maximum at approximately (2,9), decreases crossing the x-axis at approximately (4,0), continues decreasing to a minimum at approximately (6,-9), then increases again crossing the x-axis at approximately (8,0) and continuing upward. The x-axis is labeled from 0 to 10, and the y-axis shows values from -9 to 9.
    1. List all input(s) $x$ for which $f(x)=6$.
    2. On what interval(s) is the function $f(x)$ increasing?
    3. On what interval(s) is the function $f(x)$ decreasing?
  6. Let $g(x)=x^2+2x-3$.
    1. Write $g(x)$ in factored form:
    2. Does $g(x)$ open upwards or downwards? Write up or down.
    3. Which of the following three graphs could be the graph of $g(x)$?
      Graph of an upward-opening parabola on a coordinate grid with vertex below the x-axis. The function has a vertical intercept at (0,3). The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6.
      Graph of an upward-opening parabola on a coordinate grid with vertex below the x-axis. The function has a vertical intercept at (0,-3). The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6.
      Graph of a downward-opening parabola on a coordinate grid with vertex above the x-axis. The vertical intercept is at (0,-3). The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6.
  7. A gym charges new members a $\$30$ registration fee, and then $\$28$ per month.
    1. Determine the formula for the linear function $f(t)$ which gives the total amount, in dollars, you have paid if you have been a member for $t$ months.
    2. You cannot remember when you became a member, but have paid a total of $\$254$. How many months have passed since you joined?
  8. The balance (in dollars) in a savings account is described by $D(t) = 10t^2 + 3500$, where $t$ is the number of years since 2000. Find the average rate of change, including units, of $D$ between 2000 and 2016.
  9. For each of the following, write “yes” if the relationship describes a function and “no” if not.
    $x$ 1 2 3 4
    $y$ 6 3 6 2
    $x$ $y$
    1 2
    2 4
    3 2
    2 1
    Graph of a continuous curve on a coordinate plane. The curve has a wave-like shape with peaks and valleys. The x-axis and y-axis are labeled.
    Graph of a circular shape centered at the origin on a coordinate plane. The x-axis and y-axis are labeled.