Chapter 5

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=y^{2}+1$$

Solve each problem involving rate of work. A couple is laying a tile floor. Working alone, one can do the job in 20 hours. If the two of them work together, they can complete the job in 12 hours. How long would it take the other one to lay the floor working alone?

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=y^{2}-3$$

Solve each problem involving rate of work. If a vat of solution can be filled by an inlet pipe in 5 hours and emptied by an outlet pipe in 10 hours, how long will it take to fill an empty vat if both pipes are open?

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=2 y^{2}$$

Solve each problem involving rate of work. A winery has a vat to hold Merlot. An inlet pipe can fill the vat in 18 hours, and an outlet pipe can empty it in 24 hours. How long will it take to fill an empty vat if both the outlet pipe and the inlet pipe are open?

If we are given the graph of \(y=f(x)\), we can obtain the graph of \(y=-f(x)\) by reflecting across the \(x\)-axis, and we can obtain the graph of \(y=f(-x)\) by reflecting across the y-axis.You are given the graph of a rational function \(y=f(x)\). Draw \(a\) sketch by hand of the graph of (a) \(y=-f(x)\) and (b) \(y=f(-x)\) (Check your book to see graph)

Solve each problem involving rate of work. It takes an inlet pipe of a small swimming pool 20 minutes less to fill the pool than it takes an outlet pipe of the same pool to empty it. Through an error, starting with an empty pool, both pipes are left open, and the pool is filled after 4 hours. How long does it take the inlet pipe to fill the pool, and how long does it take the outlet pipe to empty it?

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=-(y+1)^{2}+2$$

Solve each problem involving rate of work. A sink can be filled by the hot-water tap alone in 4 minutes more than it takes the cold-water tap alone. If both taps are open, it takes 4 minutes, 48 seconds to fill an empty sink. How long does it take each tap individually to fill the sink?

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=(y-2)^{2}-1$$

Rational function has an oblique asymptote. Determine the equation of this asymptote. Then use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{2 x^{2}+3}{4-x} ;[-19.8,19.8] \text { by }[-50,25]$$

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x^{2}+y^{2}=100$$

Rational function has an oblique asymptote. Determine the equation of this asymptote. Then use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x^{2}+9}{x+3} ;[-13.2,13.2] \text { by }[-25,25]$$

Rational function has an oblique asymptote. Determine the equation of this asymptote. Then use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x-x^{2}}{x+2} ;[-13.2,13.2] \text { by }[-15,25]$$

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$(x-2)^{2}+y^{2}=9$$

Rational function has an oblique asymptote. Determine the equation of this asymptote. Then use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x^{2}+2 x}{1-2 x} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$(x+3)^{2}+y^{2}=16$$

\(f(x)=\frac{x^{5}+x^{4}+x^{2}+1}{x^{4}+1}\) becomes \(f(x)=x+1+\frac{x^{2}-x}{x^{4}+1}\) after the numerator is divided by the denominator. (a) What is the equation of the oblique asymptote of the graph of the function? (b) For what \(x\) -value(s) does the graph of the function intersect its asymptote? (c) As \(x \rightarrow \infty,\) does the graph of the function approach its asymptote from above or below?

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$(x+3)^{2}+(y+1)^{2}=25$$

Consider the rational function $$f(x)=\frac{x^{3}-4 x^{2}+x+6}{x^{2}+x-2}$$ Divide the numerator by the denominator and use the method of Example 3 to determine the equation of the oblique asymptote. Then determine the coordinates of the point where the graph of \(f\) intersects its oblique asymptote. Use a calculator to support your answer.

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$(x-2)^{2}+(y+4)^{2}=4$$

Use long division of polynomials to show that for $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{2}+x-12}$$ if we divide the numerator by the denominator, then the quotient polynomial is \(x^{2}-x+8,\) and the remainder is \(-20 x+100 .\) Graph both \(f(x)\) and \(g(x)=x^{2}-x+8\) in the window \([-50,50]\) by \([0,1000] .\) Comment on the appearance of the two graphs. Explain how the graph of \(f\) approaches that of \(g\) as \(|x| \rightarrow \infty\).

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x=y^{2}+6 y+9$$

Suppose a friend tells you that the graph of $$f(x)=\frac{x^{2}-25}{x+5}$$ has a vertical asymptote with equation \(x=-5 .\) Is this correct? If not, describe the behavior of the graph at \(x=-5\)

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x=y^{2}-8 y+16$$

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x=2 y^{2}+8 y+1$$

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x=-3 y^{2}-6 y+2$$