Chapter 3

Find any asymptotes of the graph of the rational function. Verify your answers by using a graphing utility to graph the function. $$f(x)=\frac{x^{2}-4 x}{x^{2}-4}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$P(x)=\frac{1-3 x}{1-x}$$

Use long division to divide. $$\left(x^{5}+7\right) \div\left(x^{3}-1\right)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{2}+6 x-2$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=-6-\frac{1}{4} x^{2}\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=(x-2)^{3}-3\)

Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{x(2+x)}{2 x-x^{2}}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(t)=\frac{1-2 t}{t}$$

Use long division to divide. $$\frac{2 x^{3}-4 x^{2}-15 x+5}{(x-1)^{2}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{2}+25$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=(x+3)^{2}-4\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=-(x-3)^{3}\)

Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{x^{2}+2 x+1}{2 x^{2}-x-3}$$

Determine whether the statement is true or false. Justify your answer. The graph of a quadratic model with a negative leading coefficient will have a maximum value at its vertex.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{1}{x+2}+2$$

Use long division to divide. $$\frac{x^{4}}{(x-1)^{3}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{2}+36$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=(x-7)^{2}+2\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=(x+4)^{3}+1\)

Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{x^{2}-16}{x^{2}+8 x}$$

Determine whether the statement is true or false. Justify your answer. The graph of a quadratic model with a positive leading coefficient will have a minimum value at its vertex.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{x^{2}}{x^{2}-4}$$

Use synthetic division to divide. $$\left(3 x^{3}-17 x^{2}+15 x-25\right) \div(x-5)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=16 x^{4}-81$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(h(x)=x^{2}-2 x+1\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=3 x^{3}-9 x+1, \quad g(x)=3 x^{3}\)

Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{3-14 x-5 x^{2}}{3+7 x+2 x^{2}}$$

Determine whether the statement is true or false. Justify your answer. Data that are positively correlated are always better modeled by a linear equation than by a quadratic equation.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{x}{x^{2}-9}$$

Use synthetic division to divide. $$\left(5 x^{3}+18 x^{2}+7 x-6\right) \div(x+3)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(y)=81 y^{4}-625$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(g(x)=x^{2}+16 x+64\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-\frac{1}{3}\left(x^{3}-3 x+2\right), \quad g(x)=-\frac{1}{3} x^{3}\)

(a) find the domain of the function, (b) decide whether the function is continuous, and (c) identify any horizontal and vertical asymptotes. Verify your answer to part (a) both graphically by using a graphing utility and numerically by creating a table of values. $$f(x)=\frac{5 x^{2}-2 x-6}{x^{2}+4}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{4(x+1)}{x(x-4)}$$

Use synthetic division to divide. $$\left(6 x^{3}+7 x^{2}-x+26\right) \div(x-3)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(z)=z^{2}-z+56$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=x^{2}-x+\frac{5}{4}\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-\left(x^{4}-4 x^{3}+16 x\right), \quad g(x)=-x^{4}\)

(a) find the domain of the function, (b) decide whether the function is continuous, and (c) identify any horizontal and vertical asymptotes. Verify your answer to part (a) both graphically by using a graphing utility and numerically by creating a table of values. $$f(x)=\frac{3 x^{2}+1}{x^{2}+x+9}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$h(x)=\frac{2}{x^{2}(x-3)}$$

Use synthetic division to divide. $$\left(2 x^{3}+14 x^{2}-20 x+7\right) \div(x+6)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$h(x)=x^{2}-4 x-3$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=x^{2}+3 x+\frac{1}{4}\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=3 x^{4}-6 x^{2}, \quad g(x)=3 x^{4}\)

(a) find the domain of the function, (b) decide whether the function is continuous, and (c) identify any horizontal and vertical asymptotes. Verify your answer to part (a) both graphically by using a graphing utility and numerically by creating a table of values. $$f(x)=\frac{x^{2}+3 x-4}{-x^{3}+27}$$

Find (a) \(f \circ g\) and (b) \(g \circ f\). $$f(x)=2 x-1, g(x)=x^{2}+3$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{3 x}{x^{2}-x-2}$$

Use synthetic division to divide. $$\left(9 x^{3}-18 x^{2}-16 x+32\right) \div(x-2)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{4}+10 x^{2}+9$$