Chapter 9

When the shot is released at an angle of \(35^{\circ},\) its height, \(y,\) in feet, can be modeled by $$y=-0.01 x^{2}+0.7 x+6.1$$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through(c) and verify your answers using the blue graph. a. What is the maximum height of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$(x+2)^{2}+x(x+1)=4$$

What is a complex number?

Your friend heard that functions are studied in intermediate and college algebra courses. He asks you what a function is. Provide him with a clear, relatively concise response.

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(y,\) in feet, can be modeled by $$y=-0.8 x^{2}+2.4 x+6$$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the equation that models the ball's parabolic path.

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$(x-1)(3 x+2)=-7(x-1)$$

What is an imaginary number?

Does \(f(x)\) mean \(f\) times \(x\) when referring to a function \(f ?\) If not, what does \(f(x)\) mean? Provide an example with your explanation.

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(y,\) in feet, can be modeled by $$y=-0.8 x^{2}+3.2 x+6$$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the equation that models the ball's parabolic path.

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$2 x^{2}-9 x-3=9-9 x$$

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(3,5) \text { and }(4,1)$$

Why is every real number also a complex number?

Explain how the vertical line test is used to determine whether a graph is a function.

You have 120 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$3 x^{2}-6 x-3=12-6 x$$

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In completing the square for \(x^{2}-7 x=5,\) we should add \(\frac{49}{4}\) to both sides.

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(1,5) \text { and }(6,2)$$

For people filing a single return, federal income tax is a function of adjusted gross income. For each value of adjusted gross income, there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits. Houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$\frac{1}{x}+\frac{1}{x+3}=\frac{1}{4}$$

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Write a perfect square trinomial whose \(x\) -term is \(-20 x\).

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-4,2) \text { and }(4,17)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The word imaginary in imaginary numbers tells me that These numbers are undefined.

What is a parabola? Describe its shape.

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$\frac{1}{x}+\frac{2}{x+3}=\frac{1}{4}$$

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(2,-2) \text { and }(5,2)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Writing \(i\) before any radical helps me to avoid placing \(i\) in the radicand.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. My height is a function of my age.

Explain how to decide whether a parabola opens upward or downward.

A football is kicked straight up from a height of 4 feet with an initial speed of 60 feet per second. The formula,$$h=-16 t^{2}+60 t+4$$ describes the ball's height above the ground, \(h\), in feet, \(t\) seconds after it is kicked. How long will it take for the football to hit the ground? Use a calculator and round to the nearest tenth of a second.

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(6,-1) \text { and }(9,5)$$

If a parabola has two \(x\) -intercepts, explain how to find them.

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-4,-1) \text { and }(2,-3)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the quadratic formula to solve a quadratic equation and \(b^{2}-4 a c\) is negative, I can be certain that the equation has two imaginary solutions.

Explain how to find a parabola's \(y\) -intercept.

Perform the indicated operations. If possible, simplify the answer. \(\frac{2 x+3}{x^{2}-7 x+12}-\frac{2}{x-3}(\) Section 7.4 , Example 7 )

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-7,-5) \text { and }(-2,-1)$$

Describe how to find a parabola's vertex.

Perform the indicated operations. If possible, simplify the answer. \(\frac{x-\frac{1}{3}}{3-\frac{1}{x}}\) (Section 7.5, Example 2 or Example 5)

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-8,-4) \text { and }(-3,-8)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The complex number \(a+0 i\) is the real number \(a\).

Solve: \(\sqrt{2 x+3}=2 x-3 .\) (Section 8.5, Example 4)

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-2 \sqrt{7}, 10) \text { and }(4 \sqrt{7}, 8)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$2+\sqrt{-4}=2-2 i$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 \pm 4 i}{2}=1 \pm 4 i$$

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 it is symmetric with respect to the \(y\) -axis

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$ for the given values of \(a, b,\) and \(c .\) Where necessary, express answers in simplified radical form. \(a=2, b=9, c=-5\)

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-\sqrt{3}, 4 \sqrt{6}) \text { and }(2 \sqrt{3}, \sqrt{6})$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A horizontal line can intersect the graph of a function at more than one point.

The length of a rectangle is 3 meters longer than the width. If the area is 36 square meters, find the rectangle's dimensions. Round to the nearest tenth of a meter.

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$ for the given values of \(a, b,\) and \(c .\) Where necessary, express answers in simplified radical form. \(a=9, b=-12, c=4\)