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2.4 Linear Functions Graphing Linear Functions Standard Form Ax + By = C Slope Average Rate of Change Linear Models

Linear Function A function  is a linear function if, for real numbers a and b,

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. Solution The x-intercept is found by letting (x) = 0 and solving for x. Add 2x; divide by 2.

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. Solution The x-intercept is 3, so we plot (3, 0). The y-intercept is Plot this point and connect the two points with a straight line. Find a check point.

Graph (x) = – 2x + 6. Give the domain and range. GRAPHING A LINEAR FUNCTION USING INTERCEPTS Example 1 Graph (x) = – 2x + 6. Give the domain and range. y Solution (0, 6) check point y-intercept (2, 2) x (3, 0) The domain and range are both(– , ). x-intercept

Graph (x) = – 3. Give the domain and range. GRAPHING A HORIZONTAL LINE Example 2 Graph (x) = – 3. Give the domain and range. y Solution Since (x), or y, always equals – 3, the value of y can never be 0. A line with no x-intercept is parallel to the x-axis. The domain is (– , ). The range is {– 3}. x Horizontal line (0, – 3)

Graph x = – 3. Give the domain and range. GRAPHING A VERTICAL LINE Example 3 Graph x = – 3. Give the domain and range. y Solution Since x always equals – 3, the value of x can never be 0, and the graph has no y-intercept and is parallel to the y-axis Vertical line ( – 3, 0) x

Graph x = – 3. Give the domain and range. GRAPHING A VERTICAL LINE Example 3 Graph x = – 3. Give the domain and range. y The domain of this relation, which is not a function, is {– 3}. The range is (– , ). Vertical line ( – 3, 0) x

Motion Problems Note In this text we will agree that if the coefficients and constant in a linear equation are rational numbers, then we will consider the standard form to be where A ≥ 0, A, B, and C are integers, and the greatest common factor of A, B, and C is 1. (If two or more integers have a greatest common factor of 1, they are said to be relatively prime.)

Graph 4x – 5y = 0. Give the domain and range. GRAPHING Ax + By = C WITH C = 0 Example 4 Graph 4x – 5y = 0. Give the domain and range. Solution Find the intercepts. Let x = 0. Let y = 0. y-intercept x-intercept This graph has one intercept-at the origin.

Graph 4x – 5y = 0. Give the domain and range. GRAPHING Ax + By = C WITH C = 0 Example 4 Graph 4x – 5y = 0. Give the domain and range. y Solution Graph the intercept (0, 0) and find another point. ( 5, 4) ( 0, 0) x

Slope An important characteristic of a straight line is its slope, a numerical measure of the steepness of a line. Geometrically it may be interpreted as the ratio of rise to run. Use two distinct points. The change in the horizontal distance, x2 – x1, is denoted as ∆x (delta x) and the change in the vertical distance, y2 – y1, is denoted as ∆y.

Slope The slope m of a line through points (x1, y1) and (x2, y2) is where ∆x ≠ 0.

Caution When using the slope formula, it makes no difference which point is used (x1, y1) or (x2, y2); however, be consistent . Start with the x- and y-values of one point (either one) and subtract the corresponding values of the other point. Be sure to write the difference of the y-values in the numerator and the difference of the x-values in the denominator.

Undefined Slope The slope of a vertical line is undefined.

Find the slope of the line through the given points. FINDING SLOPES WITH THE SLOPE FORMULA Example 5 Find the slope of the line through the given points. a. Solution Let x1 = – 4, y1 = 8, and x2 = 2, y2 = – 3. Then,

Find the slope of the line through the given points. FINDING SLOPES WITH THE SLOPE FORMULA Example 5 Find the slope of the line through the given points. b. Solution Undefined The slope of a vertical line is undefined.

Find the slope of the line through the given points. FINDING SLOPES WITH THE SLOPE FORMULA Example 5 Find the slope of the line through the given points. c. Solution Drawing a graph through these two points would produce a horizontal line.

Zero Slope The slope of a horizontal line is 0.

Find the slope of the line. FINDING THE SLOPE FROM AN EQUATION Eample 6 Find the slope of the line. Solution Find any two ordered pairs that are solutions of the equation. and

Graph the line passing through GRAPHING A LINE USING A POINT AND THE SLOPE Example 7 Graph the line passing through Solution Locate the point and move 5 units down and three units horizontally to the right. This gives a second point (2, 0) which can be used to complete the graph.

GRAPHING A LINE USING A POINT AND THE SLOPE Example 7 y (– 1, 5) Down 5 (2, 0) x Right 3

Slopes A line with a positive slope rises from left to right. A line with a negative slope falls from left to right. When the slope is positive, the function is increasing. When the slope is negative, the function is decreasing.

Average Rate of Change We know that the slope of a line is the ratio of the vertical change in y to the horizontal change in x. So, the slope gives the rate of change in y per unit of change in x, where the value of y depends on the value of x. If  is a linear function defined on [a, b], then

INTERPRETING SLOPE AS AVERAGE RATE OF CHANGE Example 8 In 2001, sales of DVD players numbered 12.7 million. In 2006, estimated sales of DVD players were 19.8 million. Find the average rate of change in DVD players, in millions, per year. Solution If x = 2001 with y = 12.7 and x = 2006 with y = 19.8, then the ordered pairs are

Solution INTERPRETING SLOPE AS AVERAGE RATE OF CHANGE Example 8 In 2001, sales of DVD players numbered 12.7 million. In 2006, estimated sales of DVD players were 19.8 million. Find the average rate of change in DVD players, in millions, per year. Solution The line through the ordered pair rises from left to right and therefore has positive slope. The sales of DVD players increased by an average of 1.42 million each year from 2001 to 2006.

Linear Models A linear cost function has the form where x represents the number of items produced, m represents the variable cost per item, and b represents the fixed costs. The fixed costs do not change as more items are made. The variable cost per item increases as more product is made. The revenue function for selling depends on the price per item p and the number of items sold x.

Linear Models A linear cost function has the form The revenue function has the form Profit is described by the profit function defined as

WRITING LINEAR COST, REVENUE, AND PROFIT FUNCTIONS Example 9 Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed cost is $1500, the variable cost per item is $100, and the item sells for $125. Write linear functions to model a. cost Solution

WRITING LINEAR COST, REVENUE, AND PROFIT FUNCTIONS Example 9 Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed costs is $1500, the variable cost per item is $100, and the item sells for $125. Write linear functions to model b. revenue Solution

WRITING LINEAR COST, REVENUE, AND PROFIT FUNCTIONS Example 9 Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed costs is $1500, the variable cost per item is $100, and the item sells for $125. Write linear functions to model c. profit Use parentheses here. Solution

d. How many items must be sold for the company to make a profit? WRITING LINEAR COST, REVENUE, AND PROFIT FUNCTIONS Example 9 Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed costs is $1500, the variable cost per item is $100, and the item sells for $125. d. How many items must be sold for the company to make a profit? Solution To make a profit, P(x) must be positive.

Solution To make a profit, P(x) must be positive. WRITING LINEAR COST, REVENUE, AND PROFIT FUNCTIONS Example 9 Solution To make a profit, P(x) must be positive. Add 1500 to each side. Divide by 25. Since the number of items must be a whole number, at least 61 items must be sold for the company to make a profit.