Chapter 4

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}-8 y=0 $$

If \(\mathrm{h}(x)=2(x+1)\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x)),\) what are possible expressions for \(\mathrm{f}(x)\) and for \(\mathrm{g}(x) ?\)

For the parabola whose equation is \(y=a x^{2}+b x+c,\) the equation of the axis of symmetry is \(x=\frac{-b}{2 a}\) . The turning point of the parabola lies on the axis of symmetry. Therefore its \(x\) -coordinate is \(\frac{-b}{2 a}\) . Substitute this value of \(x\) in the equation of the parabola to find the \(y\) -coordinates of the turning point. Write the coordinates of the turning point in terms of \(a, b,\) and \(c .\)

In \(23-28,\) write an equation of the direct variation described. The length of a line segment in inches, \(i,\) is directly proportional to the length of the segment in feet, \(f .\)

A candy store sells candy by the piece for 10 cents each. The amount that a customer pays for candy, \(y,\) is a function of the number of pieces purchased, \(x .\) a. Describe, in set-builder notation, the cost in cents of each purchase as a function of the number of pieces of candy purchased. b. Yesterday, no customer purchased more than 8 pieces of candy. List the ordered pairs that describe possible purchases yesterday. c. What is the domain for yesterday's purchases? d. What is the range for yesterday's purchases?

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}+10 x-5 y-32=0 $$

Let \(c(x)\) represent the cost of an item, \(x,\) plus sales tax and \(d(x)\) represent the cost of an item less a discount of \(\$ 10 .\) a. Write \(c(x)\) using an 8\(\%\) sales tax. b. Write \(\mathrm{d}(x)\) c. Find \(c \circ \mathrm{d}(x)\) and \(\mathrm{d} \circ \mathrm{c}(x) .\) Does \(\mathrm{c} \circ \mathrm{d}(x)=\mathrm{d} \circ \mathrm{c}(x) ?\) If not, explain the difference between the two functions. When does it makes sense to use each function? d. Which function can be used to find the amount that must be paid for an item with a \(\$ 10\) discount and 8\(\%\) tax?

In \(23-28,\) write an equation of the direct variation described. The weight of a package of meat in grams, \(g,\) is directly proportional to the weight of the package in kilograms, \(k .\)

On a particular day, the function that converts American dollars, \(x,\) to Indian rupees, \(f(x),\) is \(\mathrm{f}(x)=0.2532 x .\) Find the inverse function that converts rupees to dollars. Verify that the functions are inverses.

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}+x-3 y-2=0 $$

The relationship between temperature and the rate at which crickets chirp can be approximated by the function \(\mathrm{n}(x)=4 x-160\) where \(\mathrm{n}\) is the number of chirps per minute and \(x\) is the temperature in degrees Fahrenheit. On a given summer day, the temperature outside between the hours of 6 A.M. and 12 P.M. can be modeled by the function \(f(x)=0.55 x^{2}+\) \(1.66 x+50\) where \(x\) is the number of hours elapsed from 6 P.M. a. Find the composite function n of. b. What is the rate of chirping at 11 A.M.?

In \(23-28,\) write an equation of the direct variation described. Water is flowing into a swimming pool at the rate of 25 gallons per minute. The number of gallons of water in the pool, \(g,\) is directly proportional to the number of minutes, \(m,\) that the pool has been filling from when it was empty.

When the function \(g(x)=x^{2}+8 x+18\) is restricted to the interval \(x \geq 2,\) the inverse is \(g^{-1}(x)=\sqrt{x-2}-4\) a. Graph g for values of \(x \geq 2 .\) Graph \(\mathrm{g}^{-1}\) on the same set of axes. b. What is the domain of \(\mathrm{g}\) ? What is the range of \(\mathrm{g}\) ? c. What is the domain of \(\mathrm{g}^{-1}\) ? What is the range of \(\mathrm{g}^{-1}\) ? d. Describe the relationship between the domain and range of \(\mathrm{g}\) and its inverse.

An architect is planning the entryway into a courtyard as an arch in the shape of a semi-circle with a radius of 8 feet. The equation of the arch can be written as \((x-8)^{2}+y^{2}=64\) when the domain is the set of non-negative real numbers less than or equal to 16 and the range is the set of positive real numbers less than or equal to \(8 .\) a. Draw the arch on graph paper. b. Can a box in the shape of a cube edges measure 6 feet be moved through the arch? c. Can a box that is a rectangular solid that measures 8 feet by 8 feet be moved through the arch?

In \(23-28,\) write an equation of the direct variation described. At \(9 : 00\) A.M., Christina began to add water to a swimming pool at the rate of 25 gallons per minute. When she began, the pool contained 80 gallons of water. Christina stopped adding water to the pool at \(4 : 00\) P.M. Let \(g\) be the number of gallons of water in the pool and \(t\) be the number of minutes that have past since 9\(\cdot 00\) A.M. a. Write an equation for \(g\) as a function of \(t .\) b. What is the domain of the function? c. What is the range of the function? d. Is the function one-to-one? e. Is the function an example of direct variation? Explain why or why not.

What is the measure of a side of a square that can be drawn with its vertices on a circle of radius 10\(?\)

What are dimensions of a rectangle whose length is twice the width if the vertices are on a circle of radius 10\(?\)

Airplane passengers have been surprised to look down over farmland and see designs, called crop circles, cut into cornfields. Suppose a farmer wishes to make a design consisting of three concentric circles, that is, circles with the same center but different radiu. Write the equations of three concentric circles centered at a point in a cornfield with coordinates \((2,2) .\)