Chapter 2

When using transformations with both vertical scaling and vertical shifts, the order in which you perform the transformations matters. Let \(f(x)=|x|\). (a) Find the function \(g(x)\) whose graph is obtained by first vertically stretching \(f(x)\) by a factor of 2 and then shifting the result upward by 3 units. A table of values and/or a sketch of the graph will be helpful. (b) Find the function \(g(x)\) whose graph is obtained by first shifting \(f(x)\) upward by 3 units and then multiplying the result by a factor of 2 A table of values and/or a sketch of the graph will be helpful. (c) Compare your answers to parts (a) and (b). Explain why they are different.

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{1}{x^{2}+1} ; g(x)=\frac{2 x+1}{3 x-1}$$

Solve the quadratic equation by entering the quadratic formula in the home screen of your graphing utility. (See Technology Note on page 167.) $$-0.25 x^{2}+1.14 x-2.5=0$$

The average amount of money spent on books and magazines per household in the United States can be modeled by the function \(r(t)=-0.2837 t^{2}+5.547 t+\) \(136.7 .\) Here, \(r(t)\) is in dollars and \(t\) is the number of years since \(1985 .\) The model is based on data for the years \(1985-2000 .\) According to this model, in what year(s) was the average expenditure per household for books and magazines equal to \(\$ 160 ?\) (Source: U.S. Bureau of Labor Statistics)

A child kicks a ball a distance of 9 feet. The maximum height of the ball above the ground is 3 feet. If the point at which the child kicks the ball is the origin and the flight of the ball can be approximated by a parabola, find an expression for the quadratic function that models the ball's path. Check your answer by graphing the function.

Let \(f(x)=2 x+5\) and \(g(x)=f(x+2)-4 .\) Graph both functions on the same set of coordinate axes. Describe the transformation from \(f(x)\) to \(g(x) .\) What do you observe?

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{-x+1}{2 x+3} ; g(x)=\frac{1}{x^{2}+1}$$

Solve the quadratic equation by entering the quadratic formula in the home screen of your graphing utility. (See Technology Note on page 167.) $$0.62 t^{2}-1.29 t+1.5=0$$

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(3 x-1)^{2}$$

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(-2 x+5)^{2}$$

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[3]{4 x^{2}-1}$$

Consider a parabola that opens upward and has vertex (0,4). (a) Why does the quadratic function associated with such a parabola have no real zeros? (b) Show that \(f(x)=2 x^{2}+4\) is a possible quadratic function associated with such a parabola. Is this the only possible quadratic function associated with such a parabola? Explain. (c) Find the zeros of the function \(f\) given in part (b).

This set of exercises will draw on the ideas presented in this section and your general math background. Why must we have \(a \neq 0\) in the definition of a quadratic function?

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[5]{-x^{3}+8}$$

The shape of the Gateway Arch in St. Louis, Missouri, can be approximated by a parabola. (The actual shape will be discussed in an exercise in the chapter on exponential functions.) The highest point of the arch is approximately 625 feet above the ground, and the arch has a (horizontal) span of approximately 600 feet. (Check your book to see image) (a) Set up a coordinate system with the origin at the midpoint of the base of the arch. What are the \(x\) -intercepts of the parabola, and what do they represent? (b) What do the \(x\)- and \(y\)-coordinates of the vertex of the parabola represent? (c) Write an expression for the quadratic function associated with the parabola. (Hint: Use the vertex form of a quadratic function and find the coefficient \(a .)\) (d) What is the \(y\)-coordinate of a point on the arch whose (horizontal) distance from the axis of symmetry of the parabola is 100 feet?

Name at least two features of a quadratic function that differ from those of a linear function.

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\frac{1}{2 x+5}$$

In this problem, you will explore the relationship between factoring a quadratic expression over the complex numbers and finding the zeros of the associated quadratic function. This topic will be explained in greater detail in Chapter \(3 .\) (a) Multiply \((x+i)(x-i)\) (b) What are the zeros of \(f(x)=x^{2}+1 ?\) (c) What is the relationship between your answers to parts (a) and (b)? (d) Using your answers to parts (a)-(c) as a guide, how would you factor \(x^{2}+9 ?\) (e) Using you answers to parts (a)-(d) as a guide, how would you factor \(x^{2}+c^{2},\) where \(c\) is a positive real number?

Which of the following points lie(s) on the parabola associated with the function \(f(s)=-s^{2}+6 ?\) Justify your answer. (a) (3,-1) (b) (0,6) (c) (2,1)

We know that \(i^{2}=-1,\) but is there a complex number \(z\) such that \(z^{2}=\) i? We answer that question in this exercise. (a) Calculate \(\left(\frac{\sqrt{2}}{2}(1+i)\right)\left(\frac{\sqrt{2}}{2}(1+i)\right)\) (b) Use your answer in part (a) to find a complex number \(z\) such that \(z^{2}=i\)

The following table gives the average hotel room rate for selected years from 1990 to \(1999 .\) (Source:American Hotel and Motel Association) $$\begin{array}{cc}\text { Year } & \text { Rate (in dollars) } \\\\\hline 1990 & 57.96 \\\1992 & 58.91 \\\1994 & 62.86 \\\1996 & 70.93 \\\1998 & 78.62 \\\1999 & 81.33\end{array}$$ (a) What general trend do you notice in these figures? (b) Fit both a linear and a quadratic function to this set of points, using the number of years since 1990 as the independent variable. (c) Based on your answer to part (b), which function would you use to model this set of data, and why? (d) Using the quadratic model, find the year in which the average hotel room rate will be \(\$ 85\)

Suppose that the vertex and an \(x\) -interceptl of the parabola associated with a certain quadratic function are given by (-1,2) and \((4,0),\) respectively. (a) Find the other \(x\) -intercept. (b) Find the equation of the parabola. (c) Check your answer by graphing the function.

Examine the following table of values of a quadratic function. $$\begin{array}{cc}x & f(x) \\\\-2 & 9 \\\\-1 & 3 \\\0 & 1 \\\1 & 3 \\\2 & 9\end{array}$$ (a) What is the equation of the axis of symmetry of the associated parabola? Explain how you got your answer. (b) Find the minimum or maximum value of the function and the value of \(x\) at which it occurs. (c) Sketch a graph of the function from the values given in the table. (d) Does this function have real or nonreal zeros? Explain.

The range of a quadratic function \(g(x)=a x^{2}+b x+c\) is given by \((-\infty, 2] .\) Is \(a\) positive or negative? Justify your answer.

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[3]{5 x+7}-2$$

Is it possible for a quadratic function with real coefficients to have one real zero and one nonreal zero? Explain. (Hint: Examine the quadratic formula.)

A parabola associated with a certain quadratic function \(f\) has the point (2,8) as its vertex and passes through the point \((4,0) .\) Find an expression for \(f(x)\) in the form \(f(x)=a(x-h)^{2}+k\) (a) From the given information, find the values of \(h\) and \(k\) (b) Substitute the values you found for \(h\) and \(k\) into the expression \(f(x)=a(x-h)^{2}+k\) (c) Now find \(a\). To do this, use the fact that the parabola passes through the point \((4,0) .\) That is, \(f(4)=0\) You should get an equation having just \(a\) as a variable. Solve for \(a\) (d) Substitute the value you found for \(a\) into the expression you found in part (b). (e) Graph the function using a graphing utility and check your answer. Is (2,8) the vertex of the parabola? Does the parabola pass through (4,0)\(?\)

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=4(2 x+9)^{5}-(2 x+9)^{8}$$

Is it possible for a quadratic function to have the set of all real numbers as its range? Explain. (Hint: Examine the graph of a general quadratic function.)

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(3 x-7)^{10}+5(3 x-7)^{2}$$

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Evaluate \((f \circ f)(-1)\)

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Evaluate \((g \circ g)\left(\frac{2}{3}\right)\)

Can you write down an expression for a quadratic function whose \(x\) -intercepts are given by (2,0) and (3,0)\(?\) Is there more than one possible answer? Explain.

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((f \circ f)(t),\) and give the domain of \(f \circ f\).

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((g \circ g)(x),\) and give the domain of \(g \circ g\).

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Evaluate \((f \circ f)(2)\)

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Evaluate \((g \circ g)\left(\frac{1}{2}\right)\)

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((f \circ f)(t)\), and give the domain of \(f \circ f\)

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((g \circ g)(x),\) and give the domain of \(g \circ g\)

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=3 x-1$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=-2 x+3$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=-x^{2}+x$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=3 x^{2}+2 x$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=\frac{1}{x-3}, x \neq 3$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=\frac{1}{x+1}, x \neq-1$$

The Washington Redskins' revenue can be modeled by the function \(R(t)=245+40 t,\) where \(t\) is the number of years since 2003 and \(R(t)\) is in millions of dollars. The team's operating costs are modeled by the function \(C(t)=170+60 t,\) where \(t\) is the number of years since 2003 and \(C(t)\) is in millions of dollars. Find the profit function \(P(t) .\) (Source: Associated Press)

The number of copies of a popular mystery writer's newest release sold at a local bookstore during each month after its release is given by \(n(x)=-5 x+100\) The price of the book during each month after its release is given by \(p(x)=-1.5 x+30 .\) Find \((n p)(3) .\) Interpret your results.

Let \(n(t)\) represent the number of students attending a review session each week, starting with the first week of school. Let \(p(t)\) represent the number of tutors scheduled to work during the review session each week. Interpret the amount \(\frac{n(t)}{p(t)}\)

The exchange rate from U.S. dollars to euros on a particular day is given by the function \(f(x)=0.82 x,\) where \(x\) is in U.S. dollars. If GlobalEx Corporation has revenue given by the function \(R(t)=40+2 t,\) where \(t\) is the number of years since 2003 and \(R(t)\) is in millions of dollars, find \((f \circ R)(t)\) and explain what it represents. (Source: www.xe.com)

The conversion of temperature units from degrees Fahrenheit to degrees Celsius is given by the equation \(C(x)=\frac{5}{9}(x-32),\) where \(x\) is given in degrees Fahrenheit. Let \(T(x)=70+4 x\) denote the temperature, in degrees Fahrenheit, in Phoenix, Arizona, on a typical July day, where \(x\) is the number of hours after 6 A.M. Assume the temperature model holds until 4 P.M. of the same day. Find \((C \circ T)(x)\) and explain what it represents.