Chapter 2

How is the standard form of a circle's equation obtained from its general form?

A cellphone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot$$\$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120\end{array}\right.$$ Plan \(B\) \(\cdot \$ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 200 \\\40+0.30(t-200) & \text { if } t>200\end{array}\right.$$ Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.

Find the value of \(y\) if the line through the two given points is to have the indicated slope. \((3, y)\) and \((1,4), m--3\)

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\operatorname{int}(x-2) $$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-2 \sqrt{x+1}-1 $$

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.

A cellphone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot$$\$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120\end{array}\right.$$ Plan \(B\) \(\cdot \$ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 200 \\\40+0.30(t-200) & \text { if } t>200\end{array}\right.$$ Simplify the algebraic expression in the second line of the piecewise function for plan \(\mathrm{B}\). Then use point-plotting to graph the function.

Write a piecewise function that models each cellphone billing plan. Then graph the function. \(\$ 50\) per month buys 400 minutes. Additional time costs \(\$ 0.30\) per minute.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=|x-2| $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x|+4 $$

Does \((x-3)^{2}+(y-5)^{2}=-25\) represent the equation of a circle? What sort of set is the graph of this equation?

Write a piecewise function that models each cellphone billing plan. Then graph the function. \(\$ 60\) per month buys 450 minutes. Additional time costs \(\$ 0.35\) per minute.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=(x-1)^{3} $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x|+3 $$

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=-\sqrt{16-x^{2}} $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x+4| $$

Use a graphing utility to graph each circle whose equation is given. $$ x^{2}+y^{2}=25 $$

If one point on a line is \((3,-1)\) and the line's slope is \(-2,\) find the \(y\) -intercept.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=4 x+4, g(x)=0.25 x-1 $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x+3| $$

Use a graphing utility to graph each circle whose equation is given. $$ (y+1)^{2}=36-(x-3)^{2} $$

If one point on a line is \((2,-6)\) and the line's slope is \(-\frac{3}{2},\) find the \(y\) -intercept.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=\frac{1}{x}+2, g(x)=\frac{1}{x-2} $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)-|x+4|-2 $$

Use a graphing utility to graph each circle whose equation is given. $$ x^{2}+10 x+y^{2}-4 y-20=0 $$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=\sqrt[3]{x}-2, g(x)=(x+2)^{3} $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)-|x+3|-2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I found the inverse of \(f(x)=5 x-4\) in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by \(5,\) so \(f^{-1}(x)=\frac{x+4}{5}\).

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)--|x+4| $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with the linear function \(f(x)=3 x+5\) and \(I\) do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\).

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)--|x+3| $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming that there is no such thing as metric crickets, I modeled the information in the first frame of the cartoon with the function $$ T(n)=\frac{n}{4}+40 $$ where \(T(n)\) is the temperature, in degrees Fahrenheit, and \(n\) is the number of cricket chirps per minute.

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)--|x+4|+1 $$

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=-|x+4|+2 $$

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\). Use this information to solve. Find and interpret \(T(20,000)\)

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)-2|x+4| $$

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\) Use this information to solve. Find and interpret \(T(50,000)\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of \((x-4)+(y+6)=25\) is a circle with radius 5 centered at \((4,-6)\)

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)-2|x+3| $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=3 x,\) then \(f^{-1}(x)=\frac{1}{3 x}\)

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)--2|x+4|+1 $$

The graph of \((x-3)^{2}+(y+5)^{2}=-36\) is a circle with radius 6 centered at \((3,-5)\)

What is the slope of a line and how is it found?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The domain of \(f\) is the same as the range of \(f^{-1}\).

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)--2|x+3|+2 $$

Show that the points \(A(1,1+d), B(3,3+d),\) and \(C(6,6+d)\) are collinear (lie along a straight line) by showing that the distance from \(A\) to \(B\) plus the distance from \(B\) to \(C\) equals the distance from \(A\) to \(C\).

Describe how to write the equation of a line if the coordinates of two points along the line are known.

If \(f(x)=3 x\) and \(g(x)=x+5,\) find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\)

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-x^{3}-3 $$

Explain how to derive the slope-intercept form of a line's equation, \(y-m x+b,\) from the point-slope form $$ y-y_{1}-m\left(x-x_{1}\right) $$