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Linear Functions

Linear Functions . Chapter1, Section3. Linear Functions. A linear function is a function that can be written in the form where a and b are constants. Notice that the exponent on the variable is a 1, hence first degree (linear).

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Linear Functions

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  1. Linear Functions Chapter1, Section3

  2. Linear Functions • A linear function is a function that can be written in the form where a and b are constants. Notice that the exponent on the variable is a 1, hence first degree (linear). EX: Which of the following equations are linear functions?

  3. Intercepts The points where the graph crosses or touches the x-axis and the y-axis are called the x-intercept and the y-intercept, respectively. Graphically these points can be seen on the axes. To find the y-intercept algebraically, set x = 0 and solve the equation for y. If the solution is b, then the point on the graph is (0, b). To find the x-intercept algebraically, set y = 0 and solve for x. If a is the solution, then the resulting point is (a, 0).

  4. Example 1 A business property is purchased with a promise to pay off a $60,000 loan plus $16,500 interest on this loan by making 60 monthly payments of $1275. The amount of money, y remaining to be paid on $76,500 is reduced by $1275 each month. This can be modeled the linear function: y = 76,500 – 1275x. Find the intercepts of this equation. Interpret the intercepts for this model. What limits should there be? Use the intercept to sketch the graph. Example 2 Find the x and y-intercepts for the equation, . What does the graph look like?

  5. Slope of a Line The slope of a line is defined as: To calculate the value of the slope, we use the formula as follows. When two points are given,

  6. Find the slope of the line through the points (-3,2) and (5,-4). What does the slope mean? Find the slope of the line joining the x and y-intercept points in the previous Example 2. Example 3

  7. The Relationship Between Orientation of a Line and its Slope m>0 m=0 m<0 m is undefined

  8. Using Example 2, y = 76,500 – 1275x. • What is the slope and y-intercept? • How does the amount owed on the loan change as the number of months increases? Example 4 Slope and y-intercept The slope of the graph of the equation y = mx + b is m and the y-intercept is b, or the point on the graph (0,b). From this form we get our linear function: f(x) = mx + b.

  9. Use the graph of the function y = 16.908x – 20.945 where x is the number of years after 1990 and y is the sales in billions of dollars. • What is the slope of the graph of the function? • What is the rate at which the sales grew during this period? Example 5 Constant Rate of Change The rate of change of the linear function y = mx + b is the constant m, the slope of the graph of the function. 200 180 160 120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Years after 1990

  10. Revenue, Cost and Profit • Revenue is the money from sales of goods or services. • Cost is the expense generated in producing those good or services. • Profit is the difference between Revenue and Cost. Its what is left over from the production and sale of goods and/or services: P(x) = R(x) – C(x) When these functions are linear the rates of change are called MARGINAL COST, MARGINAL REVENUE, AND MARGINAL PROFIT. (It’s the slopes of the functions.)

  11. Example:Marginal Revenue and Marginal Profit • A company produces and sells a product with revenue given by R(x) = 89.50x dollars per unit x and cost given by C(x) = 54.36x + 6790 dollars per unit x. • What is the marginal revenue for this product and what does it mean? • Find the profit function. • What is the marginal profit for this product and what does it mean?

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