36 Sect 2.4 Linear Functions Objective 1: Graphing Linear Functions Definition A linear function is a function in the form y = f(x) = mx + b where m and b are real numbers. If m 0, then the domain and range of the linear function are all real numbers. The graph of a linear function is a straight line. One method of graphing a linear function is by finding and plotting three points that lie on the line. We can use the intercepts for one or two of these points. Finding x- and y-intercepts Step 1. Find the x-intercept(s) by replacing f(x) by zero and solving for x. Step 2. Find the y-intercept(s) by replacing x by zero and solving for y. Graph the following and then state the domain and range: Ex. 1 f(x) = 2x + 5 To find the x-intercept, let f(x) = 0 and solve: 0 = 2x + 5 implies t hat x = 2.5, so the x-intercept is (2.5, 0). To find the y-intercept, evaluate f(0): f(0) = 2(0) + 5 = 5, so the y-intercept is (0, 5) To find a third point, we can pick x = 1 and find f(1): f(1) = 2(1) + 5 = 3, so (1, 3) is a third point on the line. Now, draw the line. Since m = 2, then the domain is (, ) and the range is (, ). Definition of Horizontal and Vertical Lines Let b and c be constants. 1. If f(x) = b then it's graph is a horizontal line. The domain is (, ) and the range is {b}. In this case, f is called the constant function. 2. If x = c, then it's graph is a vertical line. The domain is {c} and the range is (, ). Note that x = c is not a function.
37 Graph the following and then state the domain and range: Ex. 2 f(x) = 2 Since there is no variable x in the equation, we can make x whatever we want and f(x) will still be 2: x f(x) 0 2 2 2 4 2 Plotting the points and drawing a line, we get a horizontal line. Notice the y-intercept is (0, 2) and there is no x-intercept. The domain is (, ) and the range is { 2}. Ex. 3 x = 2 Since there is no variable y = f(x) in the equation, we can make y whatever we want and x will still be 2: x y 2 2 2 0 2 2 Plotting the points and drawing a line, we get a vertical line. Notice the x-intercept is (2, 0) and there is no y-intercept. The domain is {2} and the range is (, ). Note that this is not a function because of the vertical line. Definition of Standard Form: A linear equation in two variables is said to be written in standard form if it is in the form: Ax + By = C, where A, B, and C are integers, A 0, the GCF of A and B is 1, and both A and B cannot be zero.
38 Graph the following and then state the domain and range: Ex. 4 4x 3y = 12 a) To find the x-intercept, b) To find the y-intercept, replace y by 0 and solve: replace x by 0 and solve: 4x 3(0) = 12 4(0) 3y = 12 4x 0 = 12 0 3y = 12 4x = 12 3y = 12 x = 3 y = 4 So, the x-intercept is (3, 0). So, the y-intercept is (0, 4) c) Before we graph, let us find a third point: Let y = 4: 4x 3(4) = 12 4x 12 = 12 4x = 24 x = 6 Thus, our table looks like: Now, graph the line. The domain is (, ) and the range is (, ). Objective 2: Slope of linear functions. Slope Formula Definition The slope of a non-vertical line, denote by m, is a measure of steepness of a line. If (x 1, y 1 ) and (x 2, y 2 ) are two distinct points on a nonvertical line, then m = "rise" "run" x y 3 0 0 4 6 4 = Δy Δx = y 2 y 1 where x 1 x 2 The symbol is the Greek letter "delta" which means a "change in" so x means a change in x. For a vertical line, the slope is undefined. Calculate the slope of the line. Then sketch the graph: Ex. 5a The line passing through Ex. 5b The line passing through the points (3, 1) and (5, 4). the points (4, 3) and ( 2, 4).
Calculating the slope, we get: m = y 2 y 1 Now, sketch the graph: = 4 ( 1) 5 3 = 5 2. m = y 2 y 1 39 Calculating the slope, we get: = 4 3 2 4 = 1 Now, draw the graph: 6 = 1 6. Ex. 6 f(x) = 3 Ex. 7 x = 2 Since there is no x term, we Since there is no y term, we can make x anything and can make y anything and f(x) = 3. So, two points on the x = 2. So, two points on the line are ( 1, 3) and (2, 3). line are ( 2, 1) & ( 2, 4). m = y 2 y 1 3 3 = 2 ( 1) = 0 3 = 0. m = y 2 y 1 4 1 = 2 ( 2) = 3 0 Now, draw the graph: which is undefined. Now, draw the graph:
In general, all horizontal lines have slope of zero. Think of a horizontal as being level ground; it has zero steepness. A vertical line however is so steep that you cannot measure it. It s slope is undefined. In summary: m > 0 (positive) The line rises from left to right. f is increasing on (, ). m < 0 (negative) The line falls from left to right. f is decreasing on (, ). 40 x = # (vertical line) f(x) = # (horizontal line) m is undefined m = 0 Cannot climb the wall. Floor is level. This is not a function. This is the constant function. Properties of Linear functions Let f(x) = mx + b be a linear function where m and b are constants. Then: 1) The graph of y = f(x) = mx + b has a y-intercept of (0, b). 2) The graph of y = f(x) = mx + b has a slope of m which is the coefficient of x. Proof: 1) To find the y-intercept, evaluate f(0): f(0) = m(0) + b = b so the y-intercept is (0, b). 2) The y-intercept, (0, b), is one point on the line. If we let x = 1, then f(1) = m(1) + b = m + b. So, (1, m + b) is a second point on the line. Thus, the slope is equal to y 2 y 1 slope of f is the coefficient of x. = m+b b 1 0 = m 1 = m. Thus, the
These two properties mean that if we have a linear function, we can solve it for y = f(x) and then read off the slope and the y-coordinate of the y- intercept from the coefficient of x and the constant term respectively. Slope-intercept form A linear equation in two variables is said to be written in slope-intercept form if it is in the form y = f(x) = mx + b where m is the slope of the line and the point (0, b) is the y-intercept. Find the slope and y-intercept and then sketch the graph: Ex. 8 2x 6y = 12 First solve the equation for y: 2x 6y = 12 2x = 2x 6y = 2x + 12 6 6 y = 1 3 x 2 The slope is 1 3 = "rise" "run" and the y-intercept is (0, 2). Plot the point (0, 2). Then from that point rise 1 unit and run 3 units to get another point. From that new point, rise another 1 unit and run 3 more units to get the third point. Now, draw the graph. Given the graph below, find a) the slope, b) y-intercept, and c) the equation of the line: Ex. 9 Ex. 10 41
42 a) The graph cross the y-axis a) The graph cross the y-axis at 4, so the y-intercept is at 2, so the y-intercept is (0, 2). (0, 4). b) Moving from the point b) Moving from the point (0, 4) to (3, 6), we have to (0, 2) to (6, 3), we have to fall 2 units and run 3 units. So, rise 1 unit and run 6 units. So, the slope is "rise" "run" = 2 3 = 2 "rise". the slope is 3 "run" = 1 6. c) Since m = 2 3 and b = 4, c) Since m = 1 and b = 2, 6 the equation is f(x) = 2 3 x 4. the equation is f(x) = 1 6 x + 2. Objective 3: Slope as an Average Rate of Change. Average Rate of Change Let f be a linear function on [a, b]. If x 1 = a, x 2 = b, y 1 = f(a), & y 2 = f(b), then The Average Rate of Change of f on [a, b] is y 2 y 1 = f(b) f(a) b a Suppose we let f(x) be the function of how many miles one has driven in x hours. The average rate of change or the slope of this function would be the average speed. Thus, if a person drove 225 miles in 3 hours, the average rate of change would be 225/3 = 75 miles per hour. For each graph, find and interpret the average rate of change: Ex. 11 Ex. 12. Cost of a TV Cost of a gallon of gas Year Month Two points on the graph are Two points on the graph are (2, 1000) and (6, 500). The (0, 3) and (5, 4). The slope is slope is then 500 1000 6 2 = 125. then 4 3 = 0.2. This means 5 0 This means the price of the TV the cost of a gallon of gas was decreasing by $125 per year. was rising 20 per month.
43 Objective 4: Linear Models: Cost, Revenue, and Profit Functions. If a cost function to produce x number of items is linear, then it can be modeled by the function C(x) = mx + b where m is the cost to produce one item and b is the fixed cost which does not change no matter how many items are produced. Similarly, if a revenue function to sell x number of units is linear, then it can be modeled by the function R(x) = px where p is the price per item sold. The profit function is the revenue minus the cost: P(x) = R(x) C(x). The break-even point is where the revenue equals the cost or where the profit is zero. This is the minimum number of items that must be produced and sold in order not to lose any money. Find the following: Ex. 13 Suppose that the cost function and revenue function are linear and all units are produced and sold. The fixed cost is $1800 and the variable cost is $15 per item. If the item sells for $45, find a) the cost function, b) the revenue function, c) the profit function and d) the break-even point. a) Since the fixed cost is $1800 and the variable cost is $15, then m = 15 and b = 1800. So, C(x) = mx + b = 15x + 1800. The cost function is C(x) = 15x + 1800. b) Since the item sells for $45, then p = 45. So, R(x) = px = 45x. The revenue function is R(x) = 45x. c) The profit function P(x) = R(x) C(x) = 45x (15x + 1800) = 45x 15x 1800 = 30x 1800. The profit function is P(x) = 30x 1800. d) The break-even point is where R(x) = C(x) or 45x = 15x + 1800 (solve for x) 30x = 1800 x = 60 Thus, 60 items must be produced and sold in order to break even. This means that in order to make a profit (P(x) > 0), the company must produce and sell at least 61 items. To verify that 60 items is the break-even point, we can evaluate R(60) and C(60) and they should yield the same value: C(60) = 15(60) + 1800 R(60) = 45(60) = 900 + 1800 = 2700 = 2700 Both yield a value of $2700, so 60 units is the break-even point.