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MAT 153: College Algebra and Trigonometry

Subsection 1.1 Representing Mathematical Relationships Exercises

  1. The height above ground, \(h\text{,}\) of a ball \(t\) seconds after being dropped off a 60 meter tall building is given by the equation
    \begin{equation*} h(t)=-4.9t^2+60 \end{equation*}
    and shown on the graph below.
    A smooth curve starts near the point (0, 60) on a grid and decreases gradually at first, then more steeply as t increases. By about t = 3.5, the curve reaches a height of 0. The graph shows a quantity that begins around 60 and steadily declines to 0 over the first few units of time.
    Figure 1.1.13.
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    1. How high is the ball at \(t=2\) seconds?
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    2. How long does it take the ball to hit the ground?
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    3. When is the ball at a height of 10 meters?
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    4. When will the height of the ball be greater than 10 meters?
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    5. Using the graph, estimate the height of the ball at \(t=1\) second, then use the equation for \(h(t)\) above to find the exact value.
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    6. Using the graph, estimate at what time the ball will be a height of 20 meters, then use the formula for \(h\) above to find the exact value.
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  2. Use the graph of \(p(n)\) below to answer the following questions.
    A smooth decreasing curve is drawn on a square grid. It begins high on the left at the point (0, 60), then slopes downward, becoming steeper as it moves to the right. The curve reaches the horizontal axis at about (3.5, 0). The graph shows a value that starts at 60 and drops steadily to 0 over the first several units of time.
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    1. What is \(p(1)\text{?}\)
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    2. What is \(p(0)\text{?}\)
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    3. Solve \(p(n)=-1\text{.}\)
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    4. Solve \(p(n)=0\text{.}\)
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    5. For which values of \(n\) is \(p(n)=1\text{?}\)
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    6. Solve \(p(n)\geq 2\text{.}\)
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  3. Use Desmos to graph the function \(f(x)=4x^3+10x^2-1\) and answer the following questions.
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    1. Evaluate \(f(-1)\text{.}\)
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    2. Solve \(f(x)=-1\text{.}\)
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    3. Solve \(f(x)=4\text{.}\)
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    4. Solve \(f(x) \geq -1\text{.}\)
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    5. Solve \(f(x) < 4\text{.}\)
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  4. Evaluate each function for the indicated value or expression.
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    1. \(v(t)=5t^2-3t+4\text{,}\) \(v(-3)\)
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    2. \(q(n)=-4 \cdot (-4)^{n^2}\text{,}\) \(q(-3)\)
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    3. \(b(x)=5x+3\text{,}\) \(b(-2x+4)\)
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    4. \(z(x)=\frac{x^2+1}{x+1}\text{,}\) \(z(4x)\)
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