Exam 1: Linear Equations and Functions

Graph the function below and experiment with the viewing window to obtain a complete graph, adjusting the ranges where necessary. Does the figure below show the complete graph? ​ ​ ​

A company distributes college logo sweatshirts and sells them for $50 each. The total cost function is linear, and the total cost for 100 sweatshirts is $4,480, whereas the total cost for 250 sweatshirts is $7,480. Find the equation for the total cost function where x is the number of sweatshirts produced and sold. ​

Suppose that a certain product has the following demand and supply functions, where p is the price per unit in dollars and q is the quantity produced. ​ Demand: Supply: ​ If a $10 tax per item is levied on the supplier, who passes it on to the consumer as a price increase, find the market equilibrium point after the tax. ​

Graph the function using a window that gives a complete graph. What does y approach as x gets large? ​ ​

Solve the system by elimination, or by any convenient method. ​

Thrifty rents a compact car for $33 per day, and Budget rents a similar car for $21 per day plus an initial fee of $96. Which of the following graphs depicts the number of days for which it would be cheaper to rent from Budget? ​

Residential customers who heat their homes with natural gas have their monthly bills calculated by adding a base service charge of $5.72 per month and an energy charge of 46.87 cents per hundred cubic feet. Write an equation for the monthly charge y (in dollars) in terms of x, the number of hundreds of cubic feet used. ​

The graph of is shown in the figure below. ​ ​ If the point P (with coordinates (a, b)) is on the graph, how are a and b related? ​

The equation in this problem leads to a linear equation. Solve for x. Because the solution to the linear equation may not be a solution to the original equation, be sure to check the solution in the original equation. ​ ​

Let and . Find and simplify. ​

Use a calculator to solve the equation. Round your answer to three decimal places. ​ ​

For many species of fish, the weight W is a function of the length L that can be expressed by ​ ​ Suppose that for a particular species k = 0.03 and that for this species the length (in centimeters) is a function of the number of years t the fish has been alive, and that this function is given by ​ ​ ​ Find in order to express W as a function of the age t of the fish. ​

A couple seeking to buy a home decides that a monthly payment of $800 fits their budget. Their bank's interest rate is 7.5%. The amount they can borrow A is a function of the time t in years that it will take to repay the debt. If we denote this function by , then the following table defines the function. ​ Source: Comprehensive Mortgage Payment Tables, Publication No. 492, Financial Publishing Co., Boston Choose the expression in the form which describes how long they will have to make payments if they need to borrow $103,000. ​

Solve the system by elimination, or by any convenient method. ​ ​

Using data from the Internal Revenue Service, the per capita tax burden T (in hundreds of dollars) can be described by , where t is the number of years past 1980. Find the value of T in 1990. Round your answer to the nearest dollar. ​

Solve the inequality. ​ ​ ​

A linear cost function is , where x represents the number of units produced. What is the cost of producing one more item if 100 are currently being produced? Round your answer to the nearest cent. ​

Find the slope m and y-intercept b of the line whose equation is given below. ​ ​

Find the equilibrium point for the following supply and demand functions below, where p is price per unit and q is the number of units produced and sold. ​ Demand: Supply: ​

Choose the equation of the line with the given slope and y-intercept. ​ Slope: 8 y-intercept: 6 ​