Write the equation of the circle, \(x^2+y^2+8x-2y-8=0\), in standard form. Find the coordinates of the center of the circle and find its radius. Sketch the circle.

Sketch the graph of the equation \(y^2+x=0\)

Which of the following correspondences satisfy the definition of a function?

1. The input value is one of the letters from the set \(\{\text{J, F, M, A, S, O, N, D}\}\) and the output value is a name of a month beginning with that letter.

2. The correspondence defined by Table 1.

3. The input value is the radius of a circle and the output value is the area of the circle.

Evaluate \(g(-3)\) and \(g\left(a^2\right)\) for the following functions.

1. \(g(x)=\dfrac{\sqrt{1-x}}{2}\)

2. \(g(x)=\dfrac{x+4}{x-2}\)

Eduardo is a part-time salesperson at Digitex Audio, a sound equipment store. Each week, he is paid a salary of \(\$200\) plus a commission of \(10\%\) of the amount of sales he generates that week, in dollars.

1. What are the input and output variables for this problem?

2. Express Eduardo’s pay for one week as a function of the sales he generates that week.

3. What was his pay for a week in which he generated \(\$4000\) worth of sales?

Table 3 gives the U.S. Postal Service’s rate for a first-class letter in 2013 as a function of the weight of the letter. Letters heavier than \(3.5\) ounces are classified separately, and are not listed here.

1. Identify the independent variable and the dependent variable.

2. Explain why this table represents a function.

3. What is the rate for a first-class letter weighing \(3.2\) ounces?

Figure 4 depicts the average high temperature in Fargo, North Dakota, in degrees Fahrenheit, as a function of the month of the year.

1. Let \(T(m)\) be the function by the graph, where \(m\) is the month of the year. What is \(T(\text{May})\)?

2. What are the valid input values for this function?

Graph the function \(f(x)=\dfrac{2}{3}x-2\)

Consider the graph of the function \(f\) shown in Figure 17.

1. Find \(f(-1)\), \(f(0)\), and \(f(2)\).

2. Find the domain of \(f\).

3. What are the \(x\)- and \(y\)-intercepts of the graph of \(f\)?

Which of the following equations describes \(y\) as a function of \(x\)?

1. \(2x-y=1\)

2. \(\lvert y\rvert =x\)

A ball is dropped from a height of \(100\) meters. At time \(t\), in seconds, the height of the ball from the ground is given by \(f(t)=-4.9t^2+100\). Calculate the following:

1. The average rate of change of \(f\) on the interval \([1,3]\).

2. The difference quotient \(\dfrac{f(a+h)-f(a)}{h}\)

If \(h(x)=\sqrt{3x^2-1}\), find two functions \(f\) and \(g\) such that \(h(x)=(f\circ g)(x)=f(g(x))\).

The cost of fuel for running a fleet of vehicles owned by GlobalEx Corporation is given in Table 1 in terms of the number of gallons used. However, the European brand of GlobalEx Corporation records its fuel consumption in units of liters. So Table 2 lists the equivalent quantity of fuel in gallons.

a. Find the cost of 55 gallons of fuel.

b. Find the cost of the 113.55 liters of fuel.

Compute \(\dfrac{f(x+h)-f(x)}{h}\), \(h\neq 0\), for \(f(x)=2x^2+1\).

Make a table of values of the functions \(f(x)=\lvert x \rvert\) and \(g(x)=\lvert x-2 \rvert\), for \(x=-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\). Use your table to sketch the graphs of the two functions. What are the domain and range of \(f\) and \(g\)?

Identify the basic function \(f(x)\) which is transformed to obtain \(g(x)\). Then use transformations to sketch the graphs of both \(f(x)\) and \(g(x)\).

1. \(g(x)=3\sqrt{x}\)

2. \(g(x)=-2\lvert x \rvert\)

3. \(g(x)=-2\lvert x+1 \rvert +3\)

Suppose the graph of a function \(g(x)\) is produced from the graph of \(f(x)=x^2\) by vertically compressing the graph of \(f\) by a factor of \(\dfrac{1}{3}\), then shifting it to the left by \(1\) unit, and finally shifting it downward by \(2\) units. Give an expression for \(g(x)\) and sketch the graphs of both \(f\) and \(g\).

At the end of the year, Jocelyn’s employer gives her an annual bonus of \(\$1000\) plus \(\$200\) for each year she has been employed by the company. Answer the following questions.

1. Find a linear function that relates Jocelyn’s bonus to the number of years of her employment with the company.

2. Use the function you found in part (a) to calculate the annual bonus Jocelyn will receive after she has worked for the company for \(8\) years,

3. Use the function you found in part (a) to calculate how long Jocelyn would have to work at the company for her annual bonus to amount to \(\$3200\).

4. Interpret the slope and \(y\)-intercept for this problem both verbally and graphically.

Find the linear function \(f\) whose input and output values are given in Table 1. Evaluate \(f(5)\).

Table 3 gives the value of a commercial printer at two different times after its purchase. Answer the following questions.

1. Identify the input and output values.

2. Express the value of the printer as a linear function of the number of years after its purchase.

3. Using the function found in part (b), find the original purchase price.

4. Assuming that the value of the printer is a linear function of the number of years after its purchase, when will the printer’s value reach \(\$0\)?

Table 6 gives the body weights of laboratory rats and the weights of their hearts in grams. All data points are given to five significant digits.

1. Let \(x\) denote the body weight. Make a scatter plot of the heart weight \(h\) vs. \(x\).

2. State, in words, any general observations you can make about the data.

3. Find an expression for the linear function that best fits the given data points.

4. Compare the actual heart weights with the heart weights predicted by your function. What do you observe?

5. If a rat weighs \(308\) grams, use your model to predict its heart weight,

The volume of a fixed mass of gas is directly proportional to its temperature. If the volume of a gas is 40 cc (cubic centimeters) at 25° C (degrees Celsius), find the following.

a. The variation constant k in the equation V=kT

b. The volume of the same gas at 50°C.

c. The temperature of the same gas if the volume is 30 cc.