Chapter 2

Use a graphing utility to solve the problem. If \(f(x)=\sqrt{x},\) graph \(3 f(x)\) and \(f(3 x)\) in the same viewing window. Are the graphs the same? Explain.

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-\frac{3}{4} x^{2}-x=2$$

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$g(t)=t^{2}+t+1$$

Ball Height A cannonball is fired at an angle of inclination of \(45^{\circ}\) to the horizontal with a velocity of 50 feet per second. The height \(h\) of the cannonball is given by $$h(x)=\frac{-32 x^{2}}{(50)^{2}}+x$$ where \(x\) is the horizontal distance of the cannonball from the end of the cannon. (a) How far away from the cannon should a person stand if the person wants to be directly below the cannonball when its height is maximum? (b) What is the maximum height of the cannonball?

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

Use a graphing utility to solve the problem. Graph \(f(x)=x^{3}\) and \(g(x)=(x-7)^{3} .\) How can the graph of \(g\) be described in terms of the graph of \(f ?\)

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$(x+1)^{2}=-25$$

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$f(t)=-t^{2}-1$$

A carpenter wishes to make a rain gutter with a rectangular cross-section by bending up a flat piece of metal that is 18 feet long and 20 inches wide. The top of the gutter is open. $$2$$ (a) Write an expression for the cross-sectional area in terms of \(x,\) the length of metal that is bent upward. (b) How much metal has to be bent upward to maximize the cross-sectional area? What is the maximum cross-sectional area?

Use a graphing utility to solve the problem. Graph the functions \(f(x)=|x-4|\) and \(g(x)=f(-x)=\) \(|(-x)-4| .\) What relationship do you observe between the graphs of the two functions? Do the same with \(f(x)=(x-2)^{2}\) and \(g(x)=f(-x)=((-x)-2)^{2} .\) What type of reflection of the graph of \(f(x)\) gives the graph of \(g(x)=f(-x) ?\)

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=3 x^{2}+1 ; g(x)=\frac{2}{x+5}$$

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$(x-2)^{2}=-16$$

The quadratic function $$p(x)=-0.387(x-45)^{2}+2.73(x-45)-3.89$$ gives the percentage (in decimal form) of puffin eggs that hatch during a breeding season in terms of \(x\), the sea surface temperature of the surrounding area, in degrees Fahrenheit. What is the percentage of puffin eggs that will hatch at \(49^{\circ} \mathrm{F} ?\) at \(47^{\circ} \mathrm{F} ?\)

Let \(P(x)\) represent the price of \(x\) pounds of coffee. Assuming the entire amount of coffee is taxed at \(6 \%,\) find an expression, in terms of \(P(x),\) for just the sales tax on \(x\) pounds of coffee.

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\sqrt{x+1} ; g(x)=-3 x-4$$

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that \(z+\bar{z}=2 a\) and \(z-\bar{z}=2 b i\)

Use the intersect feature of your graphing calculator to explore the real solution(s), if any, of \(x^{2}=x+k\) for \(k=0, k=-\frac{1}{4},\) and \(k=-3 .\) Also use the zero feature to explore the solution(s). Relate your observations to the quadratic formula.

The quadratic function $$p(x)=-0.387(x-45)^{2}+2.73(x-45)-3.89$$ gives the percentage (in decimal form) of puffin eggs that hatch during a breeding season in terms of \(x\), the sea surface temperature of the surrounding area, in degrees Fahrenheit. For what temperature is the percentage of hatched puffin eggs a maximum? Find the percentage of hatched eggs at this temperature.

Let \(S(x)\) represent the weekly salary of a salesperson, where \(x\) is the weekly dollar amount of sales generated. If the salesperson pays \(15 \%\) of her salary in federal taxes, express her after-tax salary in terms of \(S(x)\) Assume there are no other deductions to her salary.

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=5 x+1 ; g(x)=\sqrt{x-3}$$

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that \(z \bar{z}=a^{2}+b^{2}\)

The height of a ball after being dropped from a point 100 feet above the ground is given by \(h(t)=-16 t^{2}+100,\) where \(t\) is the time in seconds since the ball was dropped, and \(h(t)\) is in feet. (a) When will the ball be 60 feet above the ground? (b) When will the ball reach the ground? (c) For what values of \(t\) does this problem make sense (from a physical standpoint)?

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) (Source: The League of American Theaters and Producers, Inc.) (a) Use this model to estimate the attendance in the year \(1995 .\) Compare it to the actual value of 9 million. (b) Use this model to predict the attendance for the year 2006 (c) What is the vertex of the parabola associated with the function \(p\), and what does it signify in relation to this problem? (d) Would this model be suitable for predicting the attendance at Broadway shows for the year \(2025 ?\) Why or why not? (e) \(\quad\) Use a graphing utility to graph the function \(p\) What is an appropriate range of values for \(t ?\)

The production cost, in dollars, for \(x\) color brochures is \(C(x)=500+3 x .\) The fixed cost is \(\$ 500\) since that is the amount of money needed to start production even if no brochures are printed. (a) If the fixed cost is decreased by \(\$ 50,\) find the new cost function. (b) Graph both cost functions and interpret the effect of the decreased fixed cost.

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

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that the real part of \(z\) is equal to \(\frac{z+\bar{z}}{2}\)

The height of a ball after being dropped from the roof of a 200 -foot-tall building is given by \(h(t)=-16 t^{2}+200,\) where \(t\) is the time in seconds since the ball was dropped, and \(h(t)\) is in feet. (a) When will the ball be 100 feet above the ground? (b) When will the ball reach the ground? (c) For what values of \(t\) does this problem make sense (from a physical standpoint)?

A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the price per ticket is reduced by \(\$ 0.50 .\) The reduced price applies to all the tickets sold to the group. (a) Calculate the total cost for one, two, and five tickets. (b) Using your calculations in part (a) as a guide, find a quadratic function that gives the total cost of the tickets. (c) How many tickets must be sold to maximize the revenue for the bus company?

The area of a square is given by \(A(s)=s^{2}\) where \(s\) is the length of a side in inches. Compute the expression for \(A(2 s)\) and explain what it represents.

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

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that the imaginary part of \(z\) is equal to \(\frac{z-\bar{z}}{2 i}\)

A security firm currently has 5000 customers and charges \(\$ 20\) per month to monitor each customer's home for intruders. A marketing survey indicates that for each dollar the monthly fee is decreased, the firm will pick up an additional 500 customers. Let \(R(x)\) represent the revenue generated by the security firm when the monthly charge is \(x\) dollars. Find the value of \(x\) that results in the maximum monthly revenue.

The height of a ball thrown upward with a initial velocity of 30 meters per second from an initial height of \(h\) meters is given by $$s(t)=-16 t^{2}+30 t+h$$ where \(t\) is the time in seconds. (a) If \(h=0,\) how high is the ball at time \(t=1 ?\) (b) If \(h=20,\) how high is the ball at time \(t=1 ?\) (c) In terms of shifts, what is the effect of \(h\) on the function \(s(t) ?\)

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

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. Find the discriminant \(b^{2}-4 a c\)

A rectangular garden plot is to be enclosed with a fence on three of its sides and an existing wall on the fourth side. There is 45 feet of fencing material available. (a) Write an equation relating the amount of available fencing material to the lengths of the three sides that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) For each value of the length given in the following table of possible dimensions for the garden plot, fill in the value of the corresponding width. Use your expression from part (b) and compute the resulting area. What do you observe about the area of the enclosed region as the dimensions of the garden plot are varied? $$\begin{array}{ccc}\hline \begin{array}{c}\text { Length } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Width } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Total Amount of } \\\\\text { Fencing Material (feet) }\end{array} & \begin{array}{c}\text { Area } \\\\\text { (square feet) }\end{array} \\\\\hline 5 & & 45 \\\10 & & 45 \\\15 & & 45 \\\20 & & 45 \\\30 & & 45 \\\k & & 45 \\\& &\end{array}$$ (d) Write an expression for the area of the garden plot in terms of its length. (e) Find the dimensions that will yield a garden plot with an area of 145 square feet.

Let \(T(x)\) be the temperature, in degrees Celsius, of a point on a long rod located \(x\) centimeters from one end of the rod (where that end of the rod corresponds to \(x=0\) ). Temperature can be measured in kelvin (the unit of temperature for the absolute temperature scale) by adding 273 to the temperature in degrees Celsius. Let \(t(x)\) be the temperature function in kelvin, and write an expression for \(t(x)\) in terms of the function \(T(x)\).

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

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have two real zeros?

A rectangular sandbox is to be enclosed with a fence on three of its sides and a brick wall on the fourth side. If 24 feet of fencing material is available, what dimensions will yield an enclosed region with an area of 70 square feet?

The point (2,4) on the graph of \(f(x)=x^{2}\) has been shifted horizontally to the point \((-3,4) .\) Identify the shift and write a new function \(g(x)\) in terms of \(f(x)\).

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

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have one real zero?

A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available. (a) Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) Write an expression for the area of the plot in terms of its length. (d) Find the dimensions that will yield the maximum area.

When designing buildings, engineers must pay careful attention to how different factors affect the load a structure can bear. The following table gives the load in terms of the weight of concrete that can be borne when threaded rod anchors of various diameters are used to form joints. $$\begin{array}{cc} \text { Diameter (in.) } & \text { Load (1b) } \\ \hline 0.3750 & 2105 \\ 0.5000 & 3750 \\ 0.6250 & 5875 \\ 0.7500 & 8460 \\ 0.8750 & 11,500 \end{array}$$ (a) Examine the table and explain why the relationship between the diameter and the load is not linear. (b) The function $$f(x)=14,926 x^{2}+148 x-51$$ gives the load (in pounds of concrete) that can be borne when rod anchors of diameter \(x\) (in inches) are employed. Use this function to determine the load for an anchor with a diameter of 0.8 inch. (c) since the rods are drilled into the concrete, the manufacturer's specifications sheet gives the load in terms of the diameter of the drill bit. This diameter is always 0.125 inch larger than the diameter of the anchor. Write the function in part (b) in terms of the diameter of the drill bit. The loads for the drill bits will be the same as the loads for the corresponding anchors. (Hint: Examine the table of values and see if you can present the table in terms of the diameter of the drill bit.)

The point (-2,2) on the graph of \(f(x)=|x|\) has been shifted horizontally and vertically to the point (3,4) Identify the shifts and write a new function \(g(x)\) in terms of \(f(x)\).

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=|x| ; g(x)=\frac{x^{2}+3}{x^{2}-4}$$

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have no real zeros?

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions of dollars. The model is based on data for the years \(1981-2000 .\) When did the attendance reach \(\$ 12\) million? (Source: The League of American Theaters and Producers, Inc.)

A ball is thrown directly upward from ground level at time \(t=0\) ( \(t\) is in seconds). At \(t=3,\) the ball reaches its maximum distance from the ground, which is 144 feet. Assume that the distance of the ball from the ground (in feet) at time \(t\) is given by a quadratic function \(d(t) .\) Find an expression for \(d(t)\) in the form \(d(t)=a(t-h)^{2}+k\) by performing the following steps. (a) From the given information, find the values of \(h\) and \(k\) and substitute them into the expression \(d(t)=a(t-h)^{2}+k\) (b) Now find \(a\). To do this, use the fact that at time \(t=0\) the ball is at ground level. This will give you an equation having just \(a\) as a variable. Solve for \(a\) (c) Now, substitute the value you found for \(a\) into the expression you found in part (a). (d) Check your answer. Is (3,144) the vertex of the associated parabola? Does the parabola pass through (0,0)\(?\)