Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections to Previous Learning: Students should be able to write the equation of a line using a graph, slope intercept, two points, point-slope. Connections to AP*: AP Calculus Topic: Analysis of Functions Materials: Student Activity pages Teacher Notes: This lesson can be used as a review or as an assessment. Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1
Student Activity 1. Use the graph showing line b below to answer the following. Be sure to show the steps which justify EACH part of each answer. Your work will be graded based on the explanations that you give, not just based on your answer. a) What is the slope of this line? Explain how you got this answer two different ways. b) What is the y-intercept of this line? c) Explain how to use the graph to write an equation for this line in slope-intercept form. Give the equation of the line. d) Choose any two points on the graph and explain how to find the equation in slopeintercept form from those two points. e) Choose any point on the graph and the slope that you determined in part a. Write an equation in point-slope form for this line. f) Re-write your equation from part e in standard form. g) Re-write your equation from part e in slope-intercept form. h) If line b is shifted down units, what is the equation of the new line? i) If line b is shifted to the right 4 units, what is the equation of the new line? 10 8 6 4 line b 10 8 6 4-10 -8-6 -4 - - 4 6 8 10-10 -8-6 -4 - - 4 6 8 10-4 -4-6 -6-8 -8-10 -10 Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Student Activity. Use the points (, 0) and (0, -) to answer the following. (You may use the grid above to graph the points if you wish.) a) What is the slope of the line which passes through these two points? b) Write the equation of the line passing through these points. Use slope-intercept form. c) Write the equation of the line passing through these points. Use point-slope form. d) Write the equation of the line passing through these points in standard form. e) Write the equation of any line which is perpendicular to this line. f) Write the equation of a line which is parallel to this line which passes through the point (1, ). Give your answer in standard form. Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Answers: 1. a) One way is to just count on the coordinate plane. The rise is and the run is 1, so the slope is. Another way is to pick any two points on the line. I ll use (0, -1) and (1, 1). The slope 1 ( 1) is = =. 1 0 1 b) The y-intercept, from looking at where the graph crosses the y-axis, is 1. c) From the graph I counted the slope and found that it is, see part a. Then I found that the y- intercept is 1, see part b. Since the equation of the line in slope-intercept form is y = mx + b, the equation of this line is y = x 1 1 d) I ll use the points (1, 1) and (, ). From these points, find the slope which is =. 1 Next use this slope and one ordered pair. I ll use (1, 1). 1 = () + b. Solving this equation for b, I get b = 1. Substituting for the slope and y-intercept, the equation is y = x 1. e) I ll use (1, 1) and the slope is, so y 1 = (x 1). f) Re-write your equation from part e in standard form. y 1 = (x 1) y 1 = x after adding x and 1 to both sides -1(y x = -1) x y = 1 g) Re-write your equation from part e in slope-intercept form. y 1 = (x 1) y 1 = x after adding 1 to each side y = x 1 h) Since the original equation is y = x 1, a shift down units will take the y-intercept down from -1. Since -1 + - is -4, the new y-intercept is 4, and the new equation is y = x 4. i From looking at the graph, I see that the point (-4, -9) would move to the y-axis. This would make the new y-intercept (0, -9). The new equation would be y = x 9, or y = (x 4 ) 1 = x 9. Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 4
Answers 0. a) The slope is = =. 0 b) The equation of the line passing through these points is y = x. I know the y-intercept because it is the y-coordinate when x is zero, and we were given that point. c) Since we found the slope in part (a), and using (, 0), the equation is y 0 = (x ). d) Since the two points given were the intercepts, the equation can be found by just working backwards. The equation is x y = 6 since times is 6 and times is 6 or y = x 6. e) The equation of a line perpendicular to this line has to have a slope of possible equation would be of the form y = x + b. f) A line parallel to this line will have the same slope,. Using point-slope form, y = (x 1) clearing fractions and parentheses y 9 = x subtracting y and - from each side 7 = x y. The equation in standard form is x y = 7., so in general, a Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 5