Chapter 3

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)-6 x^{2}-6 x$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+3}{x+4}<0 $$

Describe in words the variation shown by the given equation. $$ z=\frac{k \sqrt{x}}{y^{2}} $$

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-1}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{4}+16 x^{2} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ x^{4}-3 x^{3}-20 x^{2}-24 x-8-0 $$

Solve the equation \(12 x^{3}+16 x^{2}-5 x-3=0\) given that \(-\frac{3}{2}\) is a root.

In Exercises \(45-48,\) give the domain and the range of each quadratic function whose graph is described. The vertex is \((-1,-2)\) and the parabola opens up.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x+2}<0 $$

Describe in words the variation shown by the given equation. $$ z=k x^{2} \sqrt{y} $$

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-2}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-x^{4}+4 x^{2} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ x^{4}-x^{3}+2 x^{2}-4 x-8-0 $$

Solve the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) given that \(-\frac{1}{3}\) is a root.

Give the domain and the range of each quadratic function whose graph is described. The vertex is (- 3, - 4) and the parabola opens down.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{-x+2}{x-4} \geq 0 $$

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x}+2$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{4}-2 x^{3}+x^{2} $$

Use the graph or the table to determine a solution of each equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation. $$ x^{3}+2 x^{2}-5 x-6=0 $$ CAN'T COPY THE GRAPH

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{-x-3}{x+2} \leq 0 $$

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x}+1$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{4}-6 x^{3}+9 x^{2} $$

Give the domain and the range of each quadratic function whose graph is described. Minimum \(-18\) at \(x--6\)

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{4-2 x}{3 x+4} \leq 0 $$

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+1}-2$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-2 x^{4}+4 x^{3} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 4 x^{4}-x^{3}+5 x^{2}-2 x-6-0 $$

In Exercises \(49-52,\) write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)-2 x^{2}\) but with the given point as the vertex. (5, 3)

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{3 x+5}{6-2 x} \geq 0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of this direct variation equation that has a positive constant of variation shows one variable increasing as the other variable decreases.

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+2}-2$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-2 x^{4}+2 x^{3} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 3 x^{4}-11 x^{3}-3 x^{2}-6 x+8-0 $$

Use the graph or the table to determine a solution of each equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation. $$ 2 x^{3}+11 x^{2}-7 x-6=0 $$ CAN'T COPY THE GRAPH

In Exercises \(49-52,\) write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)-2 x^{2}\) but with the given point as the vertex. (7, 4)

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x}{x-3}>0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When all is said and done, it seems to me that direct variation equations are special kinds of linear functions and inverse variation equations are special kinds of rational functions.

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{(x+2)^{2}}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=6 x^{3}-9 x-x^{5} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 2 x^{5}+7 x^{4}-18 x^{2}-8 x+8-0 $$

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{(x+1)^{2}}$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{x}>0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the language of variation, I can now state the formula for the area of a trapezoid, \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right),\) as, "A trapezoid's area varies jointly with its height and the sum of its bases"

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=6 x-x^{3}-x^{5} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 4 x^{5}+12 x^{4}-41 x^{3}-99 x^{2}+10 x+24-0 $$

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-4$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{(x+4)(x-1)}{x+2} \leq 0 $$

In a hurricane, the wind pressure varies directly as the square of the wind velocity. If wind pressure is a measure of a hurricane's destructive capacity, what happens to this destructive power when the wind speed doubles?

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=3 x^{2}-x^{3} $$

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-3$$