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Chapter 0: Problem 50

Write an equation for the function that is described by the given characteristics. The shape of \(f(x)=\sqrt{x}\), but moved nine units downward and reflected in both the \(x\)-axis and the \(y\)-axis.

Short Answer

Expert verified
The equation for the function that is moved nine units downward, and reflected in both the x and y-axes is \(f(x) = -\sqrt{-x} + 9\).

Step by step solution

01

Downward shift

Starting with the function \(f(x) = \sqrt{x}\), a downward shift by nine units is represented by the function \(f(x) = \sqrt{x} - 9\). This function will look like the original, but shifted nine units downward on the graph.
02

Reflection along the x-axis

The reflection of a function across the x-axis is accomplished by taking the opposite of the original function. So, reflecting \(f(x) = \sqrt{x} - 9\) across the x-axis would result in the function \(f(x) = -\sqrt{x} + 9\). This makes the graph get flipped around the x-axis.
03

Reflection along the y-axis

Reflection about the y-axis is accomplished by replacing \(x\) with \(-x\). Thus, reflecting \(f(x) = -\sqrt{x} + 9\) across the y-axis would give the function \(f(x) = -\sqrt{-x} + 9\). Finally, this reflects the graph about the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Downward Shift in Graphs
Shifting a graph vertically is a transformation that moves the graph up or down while preserving its shape. A downward shift occurs when a specific value is subtracted from the output of the function. For instance, suppose you have the function \(f(x) = \sqrt{x}\), which graphs as a half parabola increasing to the right. If we want to shift this graph nine units downward, we adjust the function to \(f(x) = \sqrt{x} - 9\).

Visually, every point on the graph \(f(x) = \sqrt{x}\) moves straight down by nine units. This results in the same curve, but it now lies nine units lower on the y-axis than before. An important aspect to remember about vertical shifts is that they do not affect the x-values. So, if \(x=4\) resulted in \(f(x)=2\) before the shift, after the shift, \(x=4\) would result in \(f(x)=-7\).
Reflection Across the X-Axis
A reflection across the x-axis is a type of transformation that 'flips' the graph over the x-axis. To perform this transformation algebraically, you multiply the output of the function (the \(y\) value) by -1.

Starting with \(f(x) = \sqrt{x} - 9\), reflecting it across the x-axis gives us \(f(x) = -(\sqrt{x} - 9)\), which simplifies to \(f(x) = -\sqrt{x} + 9\). In this new graph, every point \(x, y\) on the original graph \(f(x) = \sqrt{x} - 9\) is transformed to \(x, -y\) on \(f(x) = -\sqrt{x} + 9\). As a result, points above the x-axis are now below, and vice versa, it's as if the graph has been turned upside down.
Reflection Across the Y-Axis
Reflection across the y-axis is another transformative action that changes the orientation of a graph. In this transformation, for every point on the graph, the x-coordinate changes its sign. Mathematically, this is achieved by substituting \(x\) with \( -x \) in the function's formula.

Taking our previous function \(f(x) = -\sqrt{x} + 9\), reflecting it across the y-axis turns it into \(f(x) = -\sqrt{-x} + 9\). This means that for any positive \(x\) value on the original graph, there will now be a corresponding point with a negative \(x\) value after the reflection. Essentially, the graph has been mirrored over the y-axis, with the left and right sides switched.

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Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x-3}{x+2} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=3 x+8 \quad & x_{1}=0, x_{2}=3 \end{array} $$

Prescription Drugs The amounts \(d\) (in billions of dollars) spent on prescription drugs in the United States from 1991 to 2002 (see figure) can be approximated by the model $$ d(t)= \begin{cases}5.0 t+37, & 1 \leq t \leq 7 \\ 18.7 t-64, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=1\) corresponding to 1991. Use this model to find the amount spent on prescription drugs in each year from 1991 to 2002 . (Source: U.S. Centers for Medicare \& Medicaid Services)

Maximum Profit The cost per unit in the production of a portable CD player is \(\$ 60\). The manufacturer charges \(\$ 90\) per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by \(\$ 0.15\) per CD player for each unit ordered in excess of 100 (for example, there would be a charge of \(\$ 87\) per CD player for an order size of 120 ). (a) The table shows the profit \(P\) (in dollars) for various numbers of units ordered, \(x\). Use the table to estimate the maximum profit. \begin{tabular}{|l|c|c|c|c|} \hline Units, \(x\) & 110 & 120 & 130 & 140 \\ \hline Profit, \(P\) & 3135 & 3240 & 3315 & 3360 \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|} \hline Units, \(x\) & 150 & 160 & 170 \\ \hline Profit, \(P\) & 3375 & 3360 & 3315 \\ \hline \end{tabular} (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x\) ? (c) If \(P\) is a function of \(x\), write the function and determine its domain.

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}
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