Chapter 6

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=-x^{2}+1 $$

Let \(f(x)=x+3, g(x)=x-5, h(x)=f(x) g(x)\), and \(j(x)=\frac{f(x)}{g(x)}\). Solve the following equations. Find all \(x\) that satisfy the equation. (a) \(h(x)=0\) (b) \(h(x)=-7\) (c) \(h(x)=-15\) (d) \(h(x)=c \quad c\) is a constant that will appear in your answer. (e) \(j\left(x^{2}\right)-2=0\) (f) \([j(x)]^{2}-1=0\) (g) \(h(x)=j(x)\)

A seat on a round-trip charter flight to Cairo costs \(\$ 720\) plus a surcharge of \(\$ 10\) for every unsold seat on the airplane. (If there are 10 seats left unsold, the airline will charge each passenger \(\$ 720+\$ 100=\$ 820\) for the flight.) The plane seats 220 travelers and only round-trip tickets are sold on the charter flights. (a) Let \(x=\) the number of unsold seats on the flight. Express the revenue received for this charter flight as a function of the number of unsold seats. (Hint: Revenue = (price + surcharge)(number of people flying).) (b) Graph the revenue function. What, practically speaking, is the domain of the function? (c) Determine the number of unsold seats that will result in the maximum revenue for the flight. What is the maximum revenue for the flight?

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=(x+3)(2 x-6) $$

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=x^{2}+5 x-6 $$

Solve for \(x\). Work as efficiently as possible. (a) \(x^{2}-7=0\) (b) \(5 x^{2}=125\) (c) \((x+1)^{2}=25\) (d) \(x^{2}+2 x+1=25\) (e) \((2 x+3)^{2}=9\) (f) \((3 x+1)^{2}=7\) (g) \((x+3)(x-1)=0\) (h) \((x+3)(x-1)=7\)

The height of a ball (in feet) \(t\) seconds after it is thrown is given by $$ h(t)=-16 t^{2}+32 t+48=-16(t+1)(t-3) $$ (a) Graph \(h(t)\) for the values of \(t\) for which it makes sense. Below it graph \(v(t) .\) Be sure that \(v(t)\) looks like the derivative of \(h(t)\). (b) From what height was the ball thrown? (c) What was the ball's initial velocity? Was it thrown up or down? How can you tell? (d) Was the ball's height increasing or decreasing at time \(t=2\) ? (e) At what time did the ball reach its maximum height? How high was it then? What was its velocity at that time? (f) How long was the ball in the air? (g) What is the ball's acceleration? Does this make physical sense?

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=2(x-1)^{2}-4 $$

Sketch the graphs of the following functions. Use what you know about the basic shapes plus shifting, flipping, and stretching to draw the graph without plotting lots of points. In each case tell us what "basic" function you are transforming. Label the \(x\) - and \(y\) -intercepts of each graph. Work as efficiently as possible. (a) \(f(x)=2(x-3)^{2}-5\) (b) \(g(x)=-4(x+1)^{2}+3\) (c) \(h(x)=|x+1|-3\) (d) \(j(x)=|x-3|\) (e) \(k(x)=x^{2}-3\) (f) \(l(x)=\left|x^{2}-3\right|\) (Hint: This has two sharp corners.)

We know that Revenue \(=\) (price) . (quantity). Suppose a certain company has a monopoly on a good. If the company wants to increase its revenue it can do so by raising its prices up to a certain point. However, at some point the price becomes so high that there are not enough buyers and the revenue actually goes down. Therefore, if a monopolist is attempting to maximize revenue, the monopolist must look at the demand curve. Suppose the demand curve for widgets is given by $$ p=1000-4 q $$ where \(p\) is measured in dollars and \(q\) in hundreds of items. (a) Express revenue as a function of price and determine the price that maximizes the monopolist's revenue. (b) What price(s) gives half of the maximum revenue?

For each of the quadratics, identify the \(x\) - and \(y\) -coordinates of the vertex and determine whether the vertex is the highest point on the curve or the lowest point on the curve. $$ y=-\frac{2 x^{2}+7}{4}+\frac{3 x-1}{3} $$

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=\left|-x^{2}+1\right| $$

Refer to Problem 4 for your answers to this question. (a) How many solutions are there to \(2(x-3)^{2}-5=-6 ?\) (b) How many solutions are there to \(-4(x+1)^{2}+3=-6 ?\) (c) How many solutions are there to \(|x+1|-3=-2 ?\) (d) Solve \(x^{2}-3 \geq 1\). (e) How many solutions are there to \(\left|x^{2}-3\right|=1 ?\)

Suppose that \(q\), the quantity of gas (in gallons) demanded for heating purposes, is given by \(q=m p+b\), where \(m\) and \(b\) are constants \((m\) negative and \(b\) positive) and \(p\) is the price of gas per gallon. The gas company is interested in its revenue function. (a) Explain why \(m\) is negative. (b) Express revenue as a function of price. (Note that \(m\) and \(b\) are constants, so if you express revenue in terms of \(m, b\), and \(p\), you have expressed revenue as a function of price.) (c) Graph your revenue function, labeling all intercepts. (d) From your graph, determine the price that maximizes revenue. (Your answer will be in terms of \(m\) and \(b\).) (e) Find \(R^{\prime}\) and graph it.

For Problems 5 through 8, nd a quadratic or linear function \(f(x)\) whose derivative is the line speci ed and whose graph passes through: (a) the origin, (b) the point \((0,2)\). (a) \(f^{\prime}(x)=3\) (b) \(f^{\prime}(x)=\pi\)

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=-3(x+2)^{2}+9 $$

A catering company is making elegant fruit tarts for a huge college graduation celebration. The caterer insists on high quality and will not accept shoddy- looking tarts. When there are 7 pastry chefs in the kitchen they can each turn out an average of 44 tarts per hour. The pastry kitchen is not very large; let us suppose that for each additional pastry chef put to the fruit-tart task the average number of tarts per chef decreases by 4 tarts per hour. (Assume that reducing the number of chefs will increase the average production by 4 tarts per hour, until the number of chefs has decreased to \(3 .\) At that point reducing the number of chefs no longer increases the productivity of each chef.) (a) How many chefs will yield the optimum hourly fruit-tart production? (b) What is the maximal hourly fruit-tart production? (c) How many chefs are in the kitchen if the fruit-tart production is 320 tarts per hour?

Nd a quadratic or linear function \(f(x)\) whose derivative is the line speci ed and whose graph passes through: (a) the origin, (b) the point \((0,2)\). (a) \(f^{\prime}(x)=2 x\) (b) \(f^{\prime}(x)=-2 x+8\)

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=(3-x)(x+1) $$

The function \(R(p)=35 p(75-p)\) gives revenue as a function of price, \(p\), where the price is given in dollars. (a) Find the price at which the revenue is maximum. (b) What is the maximum price?

Nd a quadratic or linear function \(f(x)\) whose derivative is the line speci ed and whose graph passes through: (a) the origin, (b) the point \((0,2)\). (a) \(f^{\prime}(x)=6 x-2\) (b) \(f^{\prime}(x)=m x+b, m \neq 0\)

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=x^{2}+\pi x+1 $$

Nd a quadratic or linear function \(f(x)\) whose derivative is the line speci ed and whose graph passes through: (a) the origin, (b) the point \((0,2)\). (a) \(f^{\prime}(x)=\pi x\) (b) \(f^{\prime}(x)=-\frac{x}{3}\)

We have stated that the graph of a parabola is symmetric about the vertical line through its vertex. The goal of this problem is to prove this assertion. Since the vertex of the parabola \(f(x)=a x^{2}+b x+c\) is at \(x=\frac{-b}{2 a}\), we must show that the graph of \(f(x)\) is symmetric about the vertical line \(x=-\frac{b}{2 a} .\) This is equivalent to showing that $$ f\left(\frac{-b}{2 a}+x\right)=f\left(\frac{-b}{2 a}-x\right) $$ for all \(x\). (To arrive at this criterion on your own, do Problem 8.) Show that if \(f(x)=a x^{2}+b x+c\), then \(f\left(\frac{-b}{2 a}+x\right)=f\left(\frac{-b}{2 a}-x\right)\).

Amelia is a production potter. If she prices her bowls at \(x\) dollars per bowl, then she can sell \(120-5 x\) bowls every week. (a) For each dollar she increases her price how many fewer bowls does she sell? (b) Express her weekly revenue as a function of the price she charges per bowl. (c) Assuming that she can produce bowls more rapidly than people buy them, how much should she charge per bowl in order to maximize her weekly revenue? (d) What is her maximum weekly revenue from bowls?

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of nonvertical lines passing through \((0, \pi)\)

Solve for the indicated variable. (Your answers will be messy and will involve lots of letters. Don't let that faze you. In each problem, begin by determining whether the equation is linear or quadratic in the indicated variable. Then solve using the appropriate technique.) a) \(Q=b(j+3)-j Q\) b) \(\lambda(1+5 \lambda)=3 \pi\) c) \(\frac{\Omega}{6}=\frac{1}{3}(\lambda+\Omega+1)-\frac{1}{3} \lambda^{2}\) d) \(\Omega=3(\lambda+\Omega+1)-2 \lambda^{2}\) e) \(R+\frac{R}{\Omega}=R+V\) f) \(x(x+y)-y z-1=-k y\) g) \(x=\frac{-y \pm \sqrt{y^{2}-4(k y-y z+1)}}{2}\)

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of parabolas with \(x\) -intercepts at \(x=0\) and \(x=3\)

Solve: (a) \(x^{4}+x^{2}=6\). (b) \(x^{4}-5 x^{2}=-6\).

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of parabolas with \(x\) -intercepts at \(x=-5\) and \(x=1\)

Solve: \(2 x^{6}+5 x^{3}-3=0\).

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of parabolas with vertex \((2,3)\)

Solve: \((x-2)^{4}-2(x-2)^{2}=-1\).

The set of parabolas passing through \((0,3)\) with a slope of 2 at \((0,3)\) Strategize. In the example above we used \(y-y_{1}=m\left(x-x_{1}\right)\) as the equation of our line, not \(y=m x+b .\) Similarly, when working with parabolas, different forms of the equation for a parabola are best suited to different situations. In Problems 17 through 21, nd the equation of the parabola with the speci cations given.

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. \(x\) -intercepts of 3 and \(-2 ;\) maximum value of 1

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. \(x\) -intercepts of \(\pi\) and \(3 \pi ; y\) -intercept of 6

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. \(x\) -intercepts of \(\pi\) and \(3 \pi ; y\) -intercept of \(-2\)

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. Vertex at \((1,5) ; y\) -intercept of 1

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. Vertex at \((-2,3)\); passing through \((1,-1)\)

Sketch the graphs of the following functions. Beneath the sketch of the function, sketch the graph of the derivative. If the graph is the graph of a quadratic, label the coordinates of the vertex of the corresponding parabola; if the graph has corners, label the coordinates of the corners. (a) \(f(x)=-(x+2)^{2}\) (b) \(f(x)=(x-2)^{2}+3\) (c) \(f(x)=(x-2)(x+4)\) (d) \(f(x)=-2(x-2)(x+4)\) (e) \(f(x)=|x+4|+2\) (f) \(f(x)=|2 x|-3\)

\text { Find the equation of the parabola through the points }(0,3),(1,0), \text { and }(2,-1) \text { . }

Find the coordinates of the vertex of the parabola passing through the points \((2,0)\), \((-1,9)\), and \((1,-5)\) Decide upon a strategy for doing this problem.