Chapter 7

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=x^{2}+10 x+9\)

Graph each linear inequality. \(2 y-x>4\)

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=x+1 \\ y=3 x-1\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((4,2)\) and \((3,4)\)

Plot the given point in a rectangular coordinate system. \((-5,0)\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(f(x)=x^{2}-2 x-8\)

Graph each linear inequality. \(y>\frac{1}{3} x\)

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=-x-1 \\ 4 x-3 y=24\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-2,1)\) and \((2,2)\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(f(x)=x^{2}+4 x-5\)

Graph each linear inequality. \(y>\frac{1}{4} x\)

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=3 x-4 \\ 2 x+y=1\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-1,3)\) and \((2,4)\)

Plot the given point in a rectangular coordinate system. \((0,-4)\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=-x^{2}+4 x-3\)

a. A student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that describes total weekly earnings. b. The student is bound by the following constraints: \- To have enough time for studies, the student can work no more than 20 hours per week. \- The tutoring center requires that each tutor spend at least three hours per week tutoring. \- The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17)\), and \((8,12)\).] e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for hours per week and working as a teacher's aide for hours per week. The maximum amount that the student can earn each week is $\$$

Graph each linear inequality. \(y \leq 3 x+2\)

In Exercises 13-24, solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+y=4 \\ y=3 x\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-2,4)\) and \((-1,-1)\)

Plot the given point in a rectangular coordinate system. \(\left(-3,-1 \frac{1}{2}\right)\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=-x^{2}+2 x+3\)

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rearprojection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \- Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \- Equipment in the factory allows for making at most 200 plasma televisions in one month. \- The cost to the manufacturer per unit is \(\$ 600\) for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000 .\) Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\), \((450,100)\), and \((450,0)\).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing rear-projection televisions each month and plasma televisions each month. The maximum monthly profit is $\$$

Graph each linear inequality. \(y \leq 2 x-1\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+y=6 \\ y=2 x\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((6,-4)\) and \((4,-2)\)

Plot the given point in a rectangular coordinate system. \(\left(-3,-1 \frac{1}{2}\right)\)

In Exercises 15-22, a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 0 \\ \hline 9 & 1 \\ \hline 16 & 1.2 \\ \hline 19 & 1.3 \\ \hline 25 & 1.4 \\ \hline \end{array} $$

Graph each linear inequality. \(y<-\frac{1}{4} x\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+3 y=8 \\ y=2 x-9\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((5,3)\) and \((5,-2)\)

Plot the given point in a rectangular coordinate system. \(\left(-2,-3 \frac{1}{5}\right)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 0.3 \\ \hline 8 & 1 \\ \hline 15 & 1.2 \\ \hline 18 & 1.3 \\ \hline 24 & 1.4 \\ \hline \end{array} $$

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

Graph each linear inequality. \(y<-\frac{1}{3} x\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-3 y=-13 \\ y=2 x+7\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((3,-4)\) and \((3,5)\)

Plot the given point in a rectangular coordinate system. \((-5,-2.5)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -3 \\ \hline 1 & 2 \\ \hline 2 & 7 \\ \hline 3 & 12 \\ \hline 4 & 17 \\ \hline \end{array} $$

A theater is presenting a program on drinking and driving for students and their parents. The proceeds will be donated to a local alcohol information center. Admission is \(\$ 2\) for parents and \(\$ 1\) for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

Graph each linear inequality. \(x \leq 2\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+3 y=5 \\ 4 x+5 y=13\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((2,0)\) and \((0,8)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 5 \\ \hline 1 & 3 \\ \hline 2 & 1 \\ \hline 3 & -1 \\ \hline 4 & -3 \\ \hline \end{array} $$

On June 24,1948 , the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was 30,000 cubic feet for an American plane and 20,000 cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity, but were subject to the following restrictions: \- No more than 44 planes could be used. \- The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \- The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000\). Total weekly costs could not exceed \(\$ 300,000\). Find the number of American and British planes that were used to maximize cargo capacity.

Graph each linear inequality. \(x \leq-4\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}y=3 x-17 \\ 2 x-y=11\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((3,0)\) and \((0,-9)\)

Plot the given point in a rectangular coordinate system. \((2.5,3.5)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline 4 & 4 \\ \hline \end{array} $$

What kinds of problems are solved using the linear programming method?