Work through the questions first, then open the answer key and explanations to check your work.
The graph of \(y=f(x)\) is shown. The curve is an increasing exponential function with a horizontal asymptote at \(y=1\). The graph crosses the y-axis at the point \((0,8)\).
What is the y-coordinate of the y-intercept of the graph?
A function is defined by \(q(m)=7m^{3}\). What is the value of \(q(4)\)?
A function is defined by \(q(m)=3m^{4}\). What is the value of \(q(2)\)?
The graph of the exponential function \(y=f(x)\) is shown.
The curve is increasing and approaches the horizontal line \(y=1\) as \(x\) decreases. The graph passes through the point \((0,8)\) and another point near \((2,29)\).
What is the y-coordinate of the y-intercept of the graph?
A wildlife research team models the number of fish in a lake using the function \(F(t)=2400(0.92)^t\), where \(t\) is the number of years after 2021 and \(F(t)\) is the estimated number of fish in the lake.
According to the model, \(F(5)\approx 1580\). Which choice best interprets this statement?
Let \(c\) be a constant, and let \(y=f(x)\) be defined by \(f(x)=\frac{c-9}{x}+6\). The graph of \(y=g(x)\) is obtained by translating the graph of \(y=f(x)\) 4 units to the right and 3 units down. Which equation defines \(g(x)\)?
Let \(b\) be a constant, and let \(y=f(x)\) be defined by \(f(x)=\frac{b-9}{x}+2\). The graph of \(y=g(x)\) is obtained by translating the graph of \(y=f(x)\) 4 units to the right and 3 units down. Which equation defines \(g(x)\)?
The area of a rectangular patio is represented by \(m(3m+5)\) square feet. One side of the patio has length \(m\) feet. Which expression represents the length, in feet, of the other side of the patio?
Let \(g(x)=|28-5x|\). If \(g(t)=7t\), what is the value of \(t\)?
A function \(h(x)\) decreases by 35% each time the value of \(x\) increases by 1. If \(h(0)=16\), which equation defines \(h\)?
Let \(p\) and \(q\) be distinct nonzero real numbers such that \(p \ne -q\). The function \(y=-4(t-p)(t+q)(t-p)^{2}(t-q)\) is graphed in the \(ty\)-plane. How many distinct \(t\)-intercepts does the graph have?
A city tracks the number of bicycles available through a rental program. The number of bicycles, \(B(t)\), after \(t\) months is modeled by \(B(t)=2400(0.8)^{t/4}\).
According to this model, what is the percent decrease in the number of bicycles every 4 months?
The y-intercept of a graph is the point where the graph crosses the y-axis, which occurs when \(x=0\). The graph crosses the y-axis at \((0,8)\), so the y-coordinate of the y-intercept is \(8\).
448
Substitute 4 for \(m\) in the function: \(q(4)=7(4^{3})\). Evaluate the exponent first: \(4^{3}=64\). Then multiply: \(7\times 64=448\). Therefore, the value of \(q(4)\) is \(448\).
48
Substitute 2 for \(m\): \(q(2)=3(2^{4})\). Evaluate the exponent first: \(2^{4}=16\). Then multiply: \(3\cdot16=48\). Therefore, the value of \(q(2)\) is 48.
The y-intercept occurs where the graph crosses the y-axis, which happens when \(x=0\). The graph passes through the point \((0,8)\), so the y-coordinate of the y-intercept is \(8\).
In 2026, the lake is expected to contain about 1,580 fish.
In the function, \(t\) represents the number of years after 2021. Therefore, \(t=5\) refers to the year 2026. The value \(F(5)\approx 1580\) means that the model predicts about 1,580 fish in the lake in 2026.
\(g(x)=\frac{c-9}{x-4}+3\)
Translating a graph 4 units to the right means replacing \(x\) with \(x-4\). This gives \(f(x-4)=\frac{c-9}{x-4}+6\).
Then translating the graph 3 units down means subtracting 3 from the entire function value:
\(g(x)=\frac{c-9}{x-4}+6-3=\frac{c-9}{x-4}+3\).
Therefore, the correct equation is \(g(x)=\frac{c-9}{x-4}+3\).
\(g(x)=\frac{b-9}{x-4}-1\)
A translation 4 units to the right replaces \(x\) with \(x-4\). A translation 3 units down subtracts 3 from the entire function value.
So, \(g(x)=f(x-4)-3\).
Substituting into the definition of \(f\) gives
\(g(x)=\frac{b-9}{x-4}+2-3=\frac{b-9}{x-4}-1\).
Therefore, the correct answer is choice C.
\(3m+5\)
The area of a rectangle equals the product of its two side lengths. The area is given as \(m(3m+5)\), and one side length is \(m\). Therefore, the other side length must be the remaining factor, \(3m+5\).
\(\frac{7}{3}\)
Substitute \(t\) into the function definition:
\(|28-5t|=7t\)
Because of the absolute value, solve two cases.
Case 1: \(28-5t=7t\)
\(28=12t\), so \(t=\frac{7}{3}\).
Check the solution:
\(g\left(\frac{7}{3}\right)=\left|28-5\left(\frac{7}{3}\right)\right|=\left|\frac{84}{3}-\frac{35}{3}\right|=\frac{49}{3}\)
and
\(7\left(\frac{7}{3}\right)=\frac{49}{3}\).
So \(\frac{7}{3}\) works.
Case 2: \(28-5t=-7t\)
\(28=-2t\), so \(t=-14\).
Check this value:
\(g(-14)=|28-5(-14)|=|98|=98\), but \(7(-14)=-98\).
Since \(98\ne -98\), \(-14\) is extraneous.
Therefore, the correct answer is \(\frac{7}{3}\).
\(h(x)=16(0.65)^x\)
A 35% decrease means the output is multiplied by \(1-0.35=0.65\) for each increase of 1 in \(x\). Since \(h(0)=16\), the initial value is the coefficient. Therefore, the function has the form \(h(x)=16(0.65)^x\).
3
The graph has a \(t\)-intercept wherever \(y=0\). Set each factor equal to 0:
\(t-p=0 \Rightarrow t=p\)
\(t+q=0 \Rightarrow t=-q\)
\(t-q=0 \Rightarrow t=q\)
The factor \((t-p)^{2}\) repeats the root \(t=p\), but it does not create a new distinct intercept.
Because \(p\) and \(q\) are distinct nonzero real numbers and \(p \ne -q\), the values \(p\), \(q\), and \(-q\) are all different. Therefore, there are 3 distinct \(t\)-intercepts.
20%
In the model \(B(t)=2400(0.8)^{t/4}\), the exponent \(t/4\) shows that the factor \(0.8\) applies every 4 months. A factor of \(0.8\) means the quantity keeps 80% of its previous value each 4-month interval.
The percent decrease is therefore \(100\%-80\%=20\%\).
So the correct answer is 20%.
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