Chapter 7

In Exercises 1-6, use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(f(x)=4^{x}\)

In Exercises 1-22, graph each linear inequality. \(x+y \geq 2\)

In Exercises 1-4, determine whether the given ordered pair is a solution of the system. \((2,3)\) \(\left\\{\begin{array}{l}x+3 y=11 \\ x-5 y=-13\end{array}\right.\)

In Exercises \(1-8\), use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(x-y=3\)

In Exercises 1-20, plot the given point in a rectangular coordinate system. \((1,4)\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(f(x)=5^{x}\)

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

Determine whether the given ordered pair is a solution of the system. \((-3,5)\) \(\left\\{\begin{array}{l}9 x+7 y=8 \\ 8 x-9 y=-69\end{array}\right.\)

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(x+y=4\)

Plot the given point in a rectangular coordinate system. \((2,5)\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(y=2^{x+1}\)

Graph each linear inequality. \(3 x-y \geq 6\)

Determine whether the given ordered pair is a solution of the system. \((2,5)\) \(\left\\{\begin{array}{r}2 x+3 y=17 \\ x+4 y=16\end{array}\right.\)

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(3 x-4 y=12\)

Plot the given point in a rectangular coordinate system. \((-2,3)\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(y=2^{x-1}\)

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

Determine whether the given ordered pair is a solution of the system. \((8,5)\) \(\left\\{\begin{array}{l}5 x-4 y=20 \\ 3 y=2 x+1\end{array}\right.\)

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(2 x-5 y=10\)

Plot the given point in a rectangular coordinate system. \((-1,4)\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(f(x)=3^{x-1}\)

In Exercises 5-8, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=x+y $$ Constraints $$ \left\\{\begin{array}{l} x \leq 6 \\ y \geq 1 \\ 2 x-y \geq-1 \end{array}\right. $$

Graph each linear inequality. \(2 x+3 y>12\)

In Exercises 5-12, solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}x+y=6 \\ x-y=2\end{array}\right.\)

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(2 x+y=6\)

Plot the given point in a rectangular coordinate system. \((-3,-5)\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(f(x)=3^{x+1}\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=3 x-2 y $$ Constraints $$ \left\\{\begin{array}{l} x \geq 1 \\ x \leq 5 \\ y \geq 2 \\ x-y \geq-3 \end{array}\right. $$

Graph each linear inequality. \(2 x-5 y<10\)

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

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(x+3 y=6\)

Plot the given point in a rectangular coordinate system. \((-4,-2)\)

In Exercises 7-8, a. Rewrite each equation in exponential form. b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting \(-2,-1,0,1\), and 2 for \(y\). \(y=\log _{4} x\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=6 x+10 y $$ Constraints $$ \left\\{\begin{array}{l} x+y \leq 12 \\ x+2 y \leq 20 \\ x \geq 0 \\ y \geq 0 \end{array}\right\\} \begin{aligned} &\text { Quadrant I and } \\ &\text { its boundary } \end{aligned} $$

Graph each linear inequality. \(5 x+3 y \leq-15\)

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}2 x-3 y=6 \\ 4 x+3 y=12\end{array}\right.\)

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(5 x=3 y-15\)

Plot the given point in a rectangular coordinate system. \((4,-1)\)

a. Rewrite each equation in exponential form. b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting \(-2,-1,0,1\), and 2 for \(y\). \(y=\log _{5} x\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=x+3 y $$ Constraints $$ \left\\{\begin{array}{l} x+y \geq 2 \\ x \leq 6 \\ y \leq 5 \\ x \geq 0 \\ y \geq 0 \end{array}\right\\} \text { Quadrant I and } $$

Graph each linear inequality. \(3 x+4 y \leq-12\)

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

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(3 x=2 y+6\)

Plot the given point in a rectangular coordinate system. \((2-2)\)

In Exercises 9-14, 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. \(y=x^{2}+8 x+7\)

Use the directions for Exercises 5-8 to solve Exercises 9-12. Objective Function $$ z=5 x-2 y $$ Constraints $$ \left\\{\begin{array}{l} 0 \leq x \leq 5 \\ 0 \leq y \leq 3 \\ x+y \geq 2 \end{array}\right. $$

Graph each linear inequality. \(2 y-3 x>6\)

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=x+5 \\ y=-x+3\end{array}\right.\)

In Exercises 9-20, 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. \((2,6)\) and \((3,5)\)

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