Chapter 9

The square of the difference between a number and 7 is 16 Find the number \((s)\)

Write a linear function, \(f(x)=m x+b,\) satisfying the following conditions: $$f(0)=7 \quad \text { and } \quad f(1)=10$$

The length of a rectangle is 2 meters longer than the width. If the area is 10 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. \(a=1, b=-2, c=-6\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that there is no real number such that when twice the number is subtracted from its square, the difference is \(-5\)

If 3 times a number is increased by 2 and this sum is squared, the result is \(49 .\) Find the number(s).

If \(f(x)=a x^{2}+b x+c\) and \(r=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a},\) find \(f(r)\) without doing any algebra and explain how you arrived at your result.

The hypotenuse of a right triangle is 4 feet long. One leg is 1 foot longer than the other. Find the lengths of the legs. Round to the nearest tenth of a foot.

If 4 times a number is decreased by 3 and this difference is squared, the result is \(9 .\) Find the number \((s)\)

A car was purchased for \(\$ 22,500\). The value of the car decreases by \(\$ 3200\) per year for the first seven years. Write a function \(V\) that describes the value of the car after \(x\) years, where \(0 \leq x \leq 7 .\) Then find and interpret \(V(3)\).

The hypotenuse of a right triangle is 6 feet long. One leg is 1 foot shorter than the other. Find the lengths of the legs. Round to the nearest tenth of a foot.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The \(x\) -coordinate of the vertex of the parabola whose equation is \(y=a x^{2}+b x+c\) is \(\frac{b}{2 a}\)

Reread Exercise \(44 .\) Use your graphing utility to illustrate the problem's solution by graphing \(y=-16 x^{2}+60 x+4\) and \(y=80\) in a \([0,4,1]\) by \([0,100,10]\) viewing rectangle. Explain how the graphs show that the ball will not reach a height of 80 feet.

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$A=\pi r^{2} \text { for } r$$

Write 0.00397 in scientific notation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a parabola has only one \(x\) -intercept, then the \(x\) -intercept is also the vertex.

Graph each equation in a rectangular coordinate system. $$y=\frac{1}{3} x-2(\text { Section } 3.4, \text { Example } 3)$$

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$a x^{2}-b=0 \text { for } x$$

Divide: \(\frac{x^{3}+7 x^{2}-2 x+3}{x-2}\)

We considered two formulas that approximate the dosage of a drug prescribed for children. $$\begin{aligned}&\text { Young's rule: } C=\frac{D A}{A+12}\\\&\text { Cowling's rule: } C=\frac{D(A+1)}{24}\end{aligned}$$.In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. At which age, to the nearest tenth of a year, do the two formulas give the same dosage?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There is no relationship between the graph of \(y=a x^{2}+b x+c\) and the number of real solutions of the equation \(a x^{2}+b x+c=0\)

Graph each equation in a rectangular coordinate system. $$2 x-3 y=6$$

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$I=\frac{k}{d^{2}} \text { for } d$$

Solve: $$\left\\{\begin{array}{l}3 x+2 y=6 \\\8 x-3 y=1\end{array}\right.$$

What is the quadratic formula and why is it useful?

Graph each equation in a rectangular coordinate system. $$x=-2$$

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$A=p(1+r)^{2} \text { for } r$$

Without going into specific details for each step, describe how the quadratic formula is derived.

Find two numbers whose sum is 200 and whose product is a maximum.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find two numbers whose sum is 200 and whose product is a maximum.

Use point plotting to graph \(y=x^{2}+4 x+3 .\) Select integers from \(-5\) to \(1,\) inclusive, for \(x\)

Explain how to solve $$x^{2}+6 x+8=0$$ using the quadratic formula,

Graph \(y=2 x^{2}-8\) and \(y=-2 x^{2}+8\) in the same rectangular coordinate system. What are the coordinates of the points of intersection?

Replace \(y\) with 0 and find the \(x\) -intercepts for the graph of \(y=x^{2}-2 x-3\)

If you are given a quadratic equation, how do you determine which method to use to solve it?

A parabola has \(x\) -intercepts at 3 and \(7,\) a \(y\) -intercept at \(-21,\) and \((5,4)\) for its vertex. Write the parabola's equation.

Replace \(x\) with 0 and find the \(y\) -intercept for the graph of \(y=x^{2}-2 x-3\)

Use the Pythagorean Theorem to solve Exercises \(67-72\). Express the answer in radical form and simplify, if possible. A baseball diamond is actually a square with 90 -foot sides. What is the distance from home plate to second base? (THE IMAGES CANNOT COPY)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because I want to solve \(25 x^{2}-49=0\) fairly quickly, I'll use the quadratic formula.

Use the Pythagorean Theorem to solve Exercises \(67-72\). Express the answer in radical form and simplify, if possible. The base of a 20 -foot ladder is 15 feet from the house. How far up the house does the ladder reach? (THE IMAGES CANNOT COPY)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The fastest way for me to solve \(x^{2}-x-2=0\) is to use the quadratic formula.

a. Use a graphing utility to graph \(y=2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic equation. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, 2 of \(y=2 x^{2}-82 x+720\) is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try Xmin \(=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum point, so try Ymin \(=-130 .\) Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

Use the formula for the area of a circle, \(A=\pi r^{2},\) to solve Exercises \(71-72\). If the area of a circle is \(36 \pi\) square inches, find its radius.

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

Use the formula for the area of a circle, \(A=\pi r^{2},\) to solve Exercises \(71-72\). If the area of a circle is \(49 \pi\) square inches, find its radius.

$$\begin{array}{l}\text { I simplified } \frac{3+2 \sqrt{3}}{2} \text { to } 3+\sqrt{3} \text { because } 2 \text { is a factor of } \\\2 \sqrt{3} \text { . }\end{array}$$., Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

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

The weight of a human fetus is modeled by the formula \(W=3 t^{2}\) where \(W\) is the weight, in grams, and \(t\) is the time, in weeks, with \(0 \leq t \leq 39 .\) Use this formula to solve Exercises \(73-74\) After how many weeks does the fetus weigh 108 grams?

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

The weight of a human fetus is modeled by the formula \(W=3 t^{2}\) where \(W\) is the weight, in grams, and \(t\) is the time, in weeks, with \(0 \leq t \leq 39 .\) Use this formula to solve Exercises \(73-74\) After how many weeks does the fetus weigh 192 grams?