Find the Slope of a Line from its Graph
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- Thomasina Lindsey
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1 Math "Determining the Equation of a Line" Objectives: * Find the slope of a line. * Solve application problems involving the slope of a line. * Write the equation of a line. The concept of slope has man applications. For eample, architects use slope when designing ramps and determining the pitch of roofs. Truckers must be aware of the slope, or grade of a road. Definitions: Ratios and Rates A ratio is a comparison of two numbers using a quotient. If a and b are two numbers, the ratio of a to b is a b Ratios that are used to compare quantities with different units are called rates. Find the Slope of a Line from its Graph Definition: Slope of a Line The slope of a line (m) is a ratio that compares the vertical change to the corresponding horizontal change as we move along the line from one point to another. Eample 1: (Finding the slope of a line from its graph) Find the slope of the line graphed below Find the Slope of a Line given Two Points Definition: Slope of a Line The slope of a line passing through points ( 1, 1 ) and ( 2, 2 ) is Eample 2: (Finding the slope of a line given two points) Find the slope of the line that passes through the given points. a) (3, 1), ( 6, 2) b) ( 3, 6), (, 8) Page: 1 Notes b Bibiana Lopez
2 College Algebra b Kaufmann and Schwitters 2.3 Find the Slope of Horizontal and Vertical Lines Eample 3: (Finding the slope of horizontal and vertical lines) Find the slope of the following lines. a) = 3 b) = Slopes of Horizontal and Vertical Lines: Horizontal lines have a slope of Vertical lines have Tpes of Slopes: Positive Slope Negative Slope Zero Slope Undefined Slope m > 0 m < 0 m = 0 Applications of Slope Eample : (Applications of slope) The slope of a staircase is defined to be the ratio of the total rise to the total run, as shown in the illustration. Find the slope of the staircase. Page: 2 Notes b Bibiana Lopez
3 College Algebra b Kaufmann and Schwitters 2.3 Eample 5: (Applications of slope) It takes a skier 25 minutes to complete the course. Find his average rate of descent in feet per minute. Determine Whether Lines are Parallel or Perpendicular Using Slope Properties of Parallel and Perpendicular Lines: If two nonvertical lines have slopes of m1 and m2; then 1: The two lines are parallel if and onl if 2: The two lines are perpendicular if and onl if Eample 6: (Determining whether lines are parallel) Determine whether the line that passes through the points (3; ) ; (; 2) is parallel to a line with a slope 2. Eample 7: (Determining whether lines are parallel or perpendicular) Refer to the graph to answer the following questions. a) Find the slopes of lines l1 and l2. Are the parallel? b) Find the slopes of lines l2 and l3. Are the perpendicular? Page: 3 Notes b Bibiana Lopez
4 College Algebra b Kaufmann and Schwitters 2.3 Use Slope-Intercept Form to Write the Equation of a Line Slope-Intercept Form The equation of the line with slope m and intercept (0, b) is Eample 8: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line with slope 5 ( and intercept 0, 1 ). Write the answer in standard form. Eample 9: (Using the slope-intercept form) Find the slope and -intercept of the line graphed below. line. Write the answer in standard form. 5 Then use the slope-intercept form to write the equation of the Eample 10: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line that has the given slope and passes through the given point. a) slope 3; and passes through ( 2, 5) b) slope 2; and passes through ( 2, 8) Page: Notes b Bibiana Lopez
5 College Algebra b Kaufmann and Schwitters 2.3 Eample 11: (Applied problem) Each turn of the handle of a pencil sharpener shaves off 0.05 inch from a 7.25 inch long pencil. a) Write a linear equation that gives the new length L of the pencil after the sharpener handle has been turned t times. b) How long is the pencil after the sharpener handle has been turned 20 times? Use the Point-Slope Form to Write the Equation of a Line Point-Slope Form: The equation of the line passing through ( 1, 1 ) and with slope m is Eample 12: (Using the point-slope form) Find an equation of the line that has slope 5 and passes trough (2, 6). Write the equation in standard form. Eample 13: (Using the point-slope form) Find an equation of the line passing through ( 2, 5) and (, 3). Write the equation in slope-intercept form. Page: 5 Notes b Bibiana Lopez
6 College Algebra b Kaufmann and Schwitters 2.3 Eample 1: (Application problems) After purchasing a new drill press, a machine shop owner had his accountant prepare a depreciation worksheet for ta purposes. a) Assuming straight-line depreciation, write an equation that gives the value (v) of the drill press after ears of use. b) Find the value of the drill press after ears of use. Use Slope as an Aid When Graphing If we know the slope and the intercept of a line, we can graph the line without constructing a table of solutions. Eample 15: (Use slope as an aid when graphing) Find the slope and -intercept of the line with the equation 3 2 = and graph the line Recognize Parallel and Perpendicular Lines Eample 16: (Using slopes to recognize parallel lines) Show that the lines represented b 2 = and 6 + = 7 are parallel. Page: 6 Notes b Bibiana Lopez
7 College Algebra b Kaufmann and Schwitters 2.3 Eample 17: (Using slopes to recognize perpendicular lines) Show that the lines represented b = 6 and 2 3 = 6 are perpendicular. Eample 18: (Using slopes to find equations of perpendicular lines) Find the equation of the line that passes through ( 6, 3) and is perpendicular to the line + 3 = 12. Write the equation in slope-intercept form. When we are asked to write the equation of a line, first we must determine what we know about the graph of the line: its slope, its -intercept, points it passes through, and so on. Then substitute the appropriate numbers into one of the following forms of a linear equation. Forms of a Linear Equation Standard Form A 0 and B 0 Slope-Intercept Form slope = m and intercept is (0, b) Point-Slope Form slope = m and the line passes through ( 1, 1 ) A Horizontal Line m = 0 and the intercept is (0, b) A Vertical Line m = undefined and the intercept is (a, 0) Page: 7 Notes b Bibiana Lopez
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Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
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What are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything