Chapter 7

Find the equation of the parabola (with vertical axis that passes through the data points shown or specified. Check your answer. $$(-1,4),(1,2),(3,8)$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$B A$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=-x^{2}+2\\\&x-y=0\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq 3^{x}\\\&y \geq 2\end{aligned}$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} 3 x+6 y-3 z &=12 \\ -x-2 y+z &=16 \\ x+y-2 z &=20 \end{aligned}$$

Find the equation of the parabola (with vertical axis that passes through the data points shown or specified. Check your answer. $$(-2,2),(0,2),(2,-6)$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$A C$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=(x-1)^{2}\\\&x-3 y=-1\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \leq\left(\frac{1}{2}\right)^{x}\\\&y \geq 4\end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}2 x-y+3 z &=1 \\\\-2 x+y-3 z &=2 \\\5 x-y+z &=2\end{aligned}$$

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\). $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\ x-y-3 z-2 w &=2 \end{aligned}$$

Find the equation of the parabola (with vertical axis that passes through the data points shown or specified. Check your answer. $$(0,1),(1,0),(2,-5)$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$B C$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&x^{2}+y^{2}=5\\\&x-y=1\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}\ln x-y \geq 1 \\\x^{2}-2 x-y \leq 1\end{array}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}-2 x-2 y+3 z &=4 \\\5 x+7 y-z &=2 \\\2 x+2 y-3 z &=-4\end{aligned}$$

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\). $$\begin{aligned} 3 x+2 y-w &=0 \\ 2 x+z+2 w &=5 \\ x+2 y-z &=-2 \\ 2 x-y+z+w &=2 \end{aligned}$$

The values in the table are from a quadratic function \(f(x)=a x^{2}+b x+c .\) Find \(a, b,\) and \(c\). $$\begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & 2.9 & 1.26 & 0.56 & 0.8 & 1.98 \end{array}$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$C B$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}x^{2}+y^{2} &=5 \\\\-3 x+4 y &=2\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \leq \log x\\\&y \geq|x-2|\end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}3 x+2 y-w &=0 \\\2 x+z+2 w &=5 \\\x+2 y-z &=-2 \\\2 x-y+z+w &=2\end{aligned}$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$A B$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}x^{2}+y^{2} &=10 \\\\-x^{2}+y &=-4\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&e^{-x}-y \leq 1\\\&x-2 y \geq 4\end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}x+2 y-z+w &=8 \\\2 x-y+2 w &=8 \\\y+3 z &=5 \\\x-z &=4\end{aligned}$$

Solve each application. The average number \(y\) of paid days off for full-time workers at medium-to- large companies after \(x\) years is listed in the table. $$\begin{array}{|l|l|l|l|}\hline x \text { (years) } & 1 & 15 & 30 \\\\\hline y \text { (days) } & 9.4 & 18.8 & 21.9\end{array}$$ (a) Determine the coefficients for \(f(x)=a x^{2}+b x+c\) so that \(f\) models these data. (b) Graph function \(f\) with the data in the viewing window \([-4,32]\) by \([8,23]\) (c) Estimate the number of paid days off after 3 years of experience. Compare it with the actual value of 11.2 days.

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$C A$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=|x-1|\\\&y=x^{2}-4\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq \frac{1}{x-1}\\\&y \leq x\end{aligned}$$

In your own words, explain what happens when you apply Cramer's rule if \(D=0\)

A company produces two models of bicycles: model \(A\) and model \(B\). Model \(A\) requires 2 hours of assembly time, and model \(B\) requires 3 hours of assembly time. The parts for model \(A\) cost \(\$ 25\) per bike; those for model \(B\) cost \(\$ 30\) per bike. If the company has a total of 34 hours of assembly time and \(\$ 365\) available per day for these two models, what is the maximum number of each model that can be made in a day and use all of the available resources?

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$A^{2}$$

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\log (x+5)\\\&y=x^{2}\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq \frac{1}{x+2}\\\&y \geq x-2\end{aligned}$$

Caltek Computer Company makes two products: computer monitors and printers. Both require time on two machines: monitors, 1 hour on machine \(A\) and 2 hours on machine \(B\); printers, 3 hours on machine \(A\) and 1 hour on machine \(B\). Both machines operate 15 hours per day. What is the maximum number of each product that can be produced per day under these conditions?

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$A^{3}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y>\frac{1}{(x-1)^{2}}\\\&y>2\end{aligned}$$

To obtain funds necessary for a planned expansion, a small company took out three loans totaling \(\$ 12,500 .\) The company was able to borrow some of the money at \(2 \% .\) It borrowed \(\$ 1000\) more than \(\frac{1}{2}\) the amount of the \(2 \%\) loan at \(3 \%\) and the rest at \(2.5 \% .\) The total annual interest was \(\$ 305 .\) How much did the company borrow at each rate?

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=e^{x+1}\\\&2 x+y=3\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l}y>\frac{1}{x^{2}} \\\y>x^{2}\end{array}$$

For any matrices \(P\) and \(Q,\) what must be true for both \(P Q\) and \(Q P\) to exist?

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\sqrt[3]{x-4}\\\&x^{2}+y^{2}=6\end{aligned}$$

Which one of the choices that follow is a description of the solution set of the following system?$$x^{2}+y^{2}<36$$ $$y

Position of a Particle Suppose that the position of a particle moving along a straight line is given by $$s(t)=a t^{2}+b t+c$$ where \(t\) is time in seconds and \(a, b,\) and \(c\) are real numbers. If \(s(0)=5, s(1)=23,\) and \(s(2)=37,\) find the equation that defines \(s(t)\). Then find \(s(8)\)

Solve each problem. Yagel's Yogurt sells three types of yogurt: nonfat, regular, and supercreamy, at three locations. Location I sells 50 gallons of nonfat, 100 gallons of regular, and 30 gallons of supercreamy each day. Location II sells 10 gallons of nonfat, 90 gallons of regular, and 50 gallons of supercreamy each day. Location III sells 60 gallons of nonfat, 120 gallons of regular, and 40 gallons of supercreamy each day. (a) Write a \(3 \times 3\) matrix that shows sales for the three locations, with the rows representing the locations. (b) The incomes per gallon for nonfat, regular, and supercreamy are \(\$ 12, \$ 10,\) and \(\$ 15,\) respectively. Write a \(3 \times 1\) matrix displaying the incomes per gallon. (c) Find a matrix product that gives the daily income at each of the three locations. (d) What is Yagel's Yogurt's total daily income from the three locations?

The solutions to the equation \(x^{3}-85 x+300=0\) were found graphically. These solutions can be found exactly by using analytic methods, as shown in the next two exercises. Use synthetic division to show that 5 is a zero of \(x^{3}-85 x+300 .\) Rewrite this polynomial by factoring out \((x-5)\)

Fill in each blank with the appropriate response. The graph of the system,$$y>x^{2}+2$$ $$x^{2}+y^{2}<16\( $$y<7$$ consists of all points \)\frac{\underline{\phantom{xx}}}{\text { (above/below) }}\( the parabola given by \)y=x^{2}+2, \frac{\underline{\phantom{xx}}}{(\text { insideloutside })}\( the circle \)x^{2}+y^{2}=16,\( and \)\frac{\underline{\phantom{xx}}}{\text { (above/below) }}\( the line \)y=7$.

Position of a Particle Suppose that the position of a particle moving along a straight line is given by $$s(t)=a t^{2}+b t+c$$ where \(t\) is time in seconds and \(a, b,\) and \(c\) are real numbers. If \(s(0)=-10, s(1)=6,\) and \(s(2)=30,\) find the equation that defines \(s(t)\). Then find \(s(10)\)

Each set of data in Exercises \(79-82\) can be modeled by $$f(x)=a x^{2}+b x+c$$ (a) Find a linear system whose solution represents values of \(a, b, a n d c\) (b) Find \(f(x)\) by using a method from this section. (c) Graph \(f\) and the data in the same viewing window. (d) Make your own prediction using \(f .\) Answers will vary. A large percentage of the U.S. population will require chronic health care in the coming decades. The average caregiving age is \(50-64,\) while the typical person needing chronic care is 85 or older. The ratio of potential caregivers to those needing chronic health care will shrink in the coming years \(x,\) as shown in the table. $$\begin{array}{|l|c|c|c|} \hline \text { Year } & 1990 & 2010 & 2030 \\ \hline \text { Ratio } & 11 & 10 & 6 \end{array}$$