Chapter 3

What is a rational inequality?

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-0.25 x^{2}+40 x$$

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(79-82 .\) Then determine the number of real zeros and the number of imaginary zeros for each function. $$ f(x)-x^{6}-64 $$

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+1}{x}$$

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\)

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-4 x^{2}+20 x+160$$

Make Sense? In Exercises \(83-86,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that \(f(-x)\) is used to explore the number of negative real zeros of a polynomial function, as well as to determine whether a function is even, odd, or neither.

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y= 5 x^{2}+40 x+600$$

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=0.01 x^{2}+0.6 x+100$$

Make Sense? In Exercises \(83-86,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a fourth-degree polynomial function with integer coefficients and zeros at 1 and \(3+\sqrt{5} .\) I'm certain that \(3+\sqrt{2}\) cannot also be a zero of this function.

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-x+1}{x-1}$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{2}+3 x-10>0 $$

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made an error when graphing this parabola because its axis of symmetry is the \(y\) -axis.

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}-1}{x^{2}-9}$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{3}+x^{2}-4 x-4>0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I like to think of a parabola's vertex as the point where it intersects its axis of symmetry.

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I threw a baseball vertically upward and its path was a parabola.

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x-5}{10 x-2} \div \frac{x^{2}-10 x+25}{25 x^{2}-1}$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{x+2}{x-3} \leq 2 $$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{1}{x+1} \leq \frac{2}{x+4} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. No quadratic functions have a range of \((-\infty, \infty)\)

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\dots+a_{1} x+a_{0}-0\) and let \(\frac{p}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as \(a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}--a_{0} q^{n}\) b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{p}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x-\frac{1}{x}}{x+\frac{1}{x}}$$

Make Sense? In Exercises \(94-97\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, 1 only pay attention to the sign of \(f\) at each test value and not the actual function value.

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x+7}{x+3}$$

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{3 x+7}{x+2}$$

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=(x-3)^{2}+2 ; \quad(6,11)$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{3 x-7}{x-2}$$

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

Explain why a polynomial function of degree 20 cannot cross the \(x\) -axis exactly once.

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x-9}{x-4}$$

Write the equation of each parabola in standard form. Vertex: \((-3,-1) ;\) The graph passes through the point \((-2,-3)\)

A company is planning to manufacture mountain bikes. The fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. a. Write the cost function, \(C\), of producing \(x\) mountain bikes. b. Write the average cost function, \(C,\) of producing x mountain bikes c. Find and interpret \(C(500), C(1000), C(2000),\) and \(C(4000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(C\) ? Describe what this means in practical terms.

In Exercises \(98-101\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3\), resulting in the equivalent inequality \(x-2<2(x+3)\)

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000 .\) It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C\), of producing \(x\) pairs of shoes. b. Write the average cost function, \(C\), of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), C(10,000),\) and \(C(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(C ?\) Describe what this represents for the company.

In Exercises \(98-101\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

The annual yield per lemon tree is fairly constant at 320 pounds when the number of trees per acre is 50 or fewer. For each additional tree over \(50,\) the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. Find the number of trees that should be planted on an acre to produce the maximum yicld. How many pounds is the maximum yield?

Each group member should consult an almanac, newspaper, magaxine, or the Internet to find data that initially increase and then decrease, or vice versa, and therefore can be modeled by a quadratic function. Group members should select the two sets of data that are most interesting and relevant. For each data set selected, a. Use the quadratic regression feature of a graphing utility to find the quadratic function that best fits the data. b. Use the equation of the quadratic function to make a prediction from the data. What circumstances might affect the ac acy of your prediction? c. Use the equation of the quadratic function to write and solve a problem involving maximizing or minimizing the function.

Write a polynomial inequality whose solution set is \([-3,5]\)

Exercises will help you prepare for the material covered in the next section. Factor: \(x^{3}+3 x^{2}-x-3\)

Write a rational inequality whose solution set is \((-\infty,-4) \cup[3, \infty)\)

Exercises will help you prepare for the material covered in the next section. If \(f(x)=x^{3}-2 x-5,\) find \(f(2)\) and \(f(3) .\) Then explain why the continuous graph of \(f\) must cross the \(x\) -axis between 2 and 3.