WARM UP EXERCSE. 1-3 Linear Functions & Straight lines

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1 WARM UP EXERCSE A company makes and sells inline skates. The price-demand function is p (x) = (x 10) 2. Describe how the graph of function p can be obtained from one of the library functions. Sketch the function Linear Functions & Straight lines The student will learn about: Straight lines Linear functions and straight lines Intercepts and graphs Slopes Special forms of lines 2

2 Graph y = -(1/3) x + 4. Examples Graph y = 3 x - 5. Graph y = (.02) x - 3. Graph y = m x +b. 3 Linear Functions And Equations. Def: A function f is a linear function if f (x) = mx + b where m and b are real numbers. The domain is the set of all real numbers, and the range is the set of all real numbers. Ex f (x) = 2x + 3 If m = 0, then f is called a constant function, which has a domain of the set of all real numbers and a range of b. Ex. f (x) = 3 The graph of a linear function is a. The graph of a constant function is a horizontal straight line. Is a vertical straight line the graph of a function? 4

3 Examples Find equations for the following graphs: 5 y = x 3 y = ax 5 1. What is m? Slope of a Line Def: If P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) are two points on a line then the slope of that line is given by the formulas: m y! = y x! x Vertical Change = = Horizontal Change Rise. Run 2. What is b? 6

4 Equations for lines b Every line can be described by a linear equation. { If the line is vertical then the equation has the form: If the line is horizontal then the equation has the form: If the line is not vertical then the equation has the form: y=mx+b linear function form or slope-intercept form All of these forms can be rewritten to the STANDARD FORM: 7 Ax + By = C Ax + By = C is the standard form of a line. A, B, and C are real constants and not both A and B are 0. Ex. Put y = -2/3 x + 4 into standard form. If A = 0 the above equation is a horizontal line with y intercept at C/B. Ex. 3y = 6 If B = 0 the above equation is a vertical line with x intercept at C/A. Ex. 2x = 6 8

5 Standard form to linear function form. 2x + 3y = 6 Write this as y=f(x) if possible. 2x = 6 Write this as y=f(x) if possible. 9 X - Intercepts The x intercept of the line defined by Ax + By = C occurs where the graph of the function crosses the x-axis (if such a point exists). -Algebraically it is the x value where -That is, to find the x intercept, let y=0 and solve for x. 2x + 3y = 6 2x = 6 The x intercept is 10

6 Y - Intercept The y intercept of the line defined by Ax + By = C occurs where the graph of the function crosses the y-axis (if such a point exists). -Algebraically it is the -That is, to find the x intercept, let x=0 and solve for y. 2x + 3y = y = 6 The y intercept is 11 Standard form to Slope Int. Form Def: The equation y = mx + b is called the slope-intercept form of a line. The slope is m and the y intercept is b. Ex. a. Put y + 2/3 x =4 into slope-intercept form. What does it tell you? b. Find the x and y intercepts to two decimal places. c. Graph the function. 12

7 Special Forms of Line Equations Def: The equation y y 1 = m (x x 1 ) is called the point-slope form of a line. The slope is m and the line passes through the point (x 1, y 1 ). Ex. Put y = -2/3 x + 4 into point-slope form for (0, ). What does it tell you? What about for (1, )? 13 Standard Review Equations of a Line Slope-Intercept Form Ax + By = C y = mx + b Point-slope form y y 1 = m (x x 1 ) Horizontal line Vertical line y = b x = a 14

8 Application Example Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly (for tax purposes) from $20,000 to $2,000: 1. Find the linear equation that relates value (V) in dollars to time (t) in years. We know two points. What are they? 15 Application Example Continued Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly for (tax purposes) from $20,000 to $2,000: With points (0, 20000) and (10, 2000) use the point-slope formula. (We could also use the slope-intercept formula on this one since we know that b = 20,000.) 16

9 Application Example Continued Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly for (tax purposes) from $20,000 to $2,000: Points are (10, 2000) and (0, 20000). The slope is!v/!t or m = You may use either point. I will choose (10, 2000). Substituting into V V 1 = m (t t 1 ), yields 17 Application Example Continued Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly for (tax purposes) from $20,000 to $2,000: Find the slope intercept form: 2. What would be the value of the equipment after 6 years? 18

10 Application Example Continued 3. Graph the equation V = -1800t ! t! Write a verbal interpretation of the slope of the line found in the first part of this problem. 19 Summary. We learned about straight lines and the different forms for straight lines. We did an applied problem involving a straight line graph and say the meaning of the y-intercept and the slope of the graph. 20