Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice

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Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice Name Date CP If an equation is linear, then there are three formats typically

Discuss in your groups what a y-intercept is. Write a brief explanation of y-intercept in your own words: 2. Draw a line with a y-intercept of 3. 3. Draw a line with a y-intercept of -2. 4.

The easiest format for writing an equation of a line given a point and the slope (or given two points from which to calculate the slope) is Point-Slope Form. (See box above.

1 Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice Name Date CP If an equation is linear, then there are three formats typically used to express the equation: y 3 = 2(x 7) y = 3x 10 3x + 2y = 12 y + 4 = -5(x 1) y = ½ x + 5 x 4y = 8 y 1 = 2 / 3 (x + 6) y = (-¼)x x = 5 y = - 3 / 4 (x + 4) y = 6 y = Discuss in your groups what a y-intercept is. Write a brief explanation of y-intercept in your own words: 2. Draw a line with a y-intercept of Draw a line with a y-intercept of In general, a y-intercept is a point on a graph that has =. 5. What is an x-intercept? 6. Draw a line with an x-intercept of Draw a line with an x-intercept of The easiest format for writing an equation of a line given a point and the slope (or given two points from which to calculate the slope) is Point-Slope Form. (See box above.) In y 1 = 2 / 3 (x + 6), the point used was (-6, 1) and the slope was 2 / 3. What is the given point and what is the given slope for the following Point-Slope equations of lines? a) y 3 = 2(x 7) Pt: m: b) y + 4 = -5(x 1) Pt: m: c) y = - 3 / 4 (x + 4) Pt: m:

2 Though it makes equation writing very simple, we do not use Point-Slope Form most of the time when writing linear equations. A primary reason not to use that format is that it could get confusing -- one particular line has an infinite number of valid equations that could be used to represent it! (We could write one new equation for each point on the line = infinite correct equations!) As such, we most often deal with linear equations in Slope-Intercept Form. After all, an individual line can have just one consistent slope throughout the entire line and it will cross the y-axis at most one time, so writing an equation that names a line s slope and y-intercept gives us a unique description of that individual line. Fortunately, it is actually simple to convert equations from Point-Slope Form into Slope-Intercept Form. All you have to do is distribute the slope into the parentheses and isolate the y, combining any like terms to the extent possible. Starting from y 1 = 2 / 3 (x + 6), we can distribute the 2 / 3, giving us y 1 = 2 / 3 x + 4. Then, add 1 to each side of the equation in order to isolate y, giving us y = 2 / 3 x + 5, the unique y = mx + b format of equation for this particular line. For problems 9 11, first graph the line passing through the given point with the given slope. Then, write an equation for the line in Point-Slope Form. Finally, convert that linear equation into its Slope-Intercept Form. 9. Slope = -1 and passes through (-4, 2) 10. Slope = 3 / 2 and passes through (-2, -4) 11. Slope = - 3 / 4 and passes through (2, 2) 12. Without graphing, find the equations of the line with slope = 5 and passing through (-10, 12).

3 For problems 13 16, rewrite the following equations in the Slope-Intercept Form if they are not already in that form. Then, state the slope and y-intercept given in each equation. 13. y = x x + y = y = 28x 16. y = 9 Slope: Slope: Slope: Slope: y-intercept: y-intercept: y-intercept: y-intercept: Standard Form is another way for us to express linear functions in an equation format. Standard Form can be a natural way to express the equation of a line, especially in real world contexts. Standard Form equations can also be great for graphing purposes because it only takes points to determine the location of a line, and it is easy to find the x-intercept and the y-intercept of a function in Standard Form. After all, the x-intercept of a graph is whatever x equals when y =, and the y-intercept of a graph is whatever y is when x =. For example, the x-intercept of 3x + 2y = 12 is (4, 0) & the y-intercept of 3x + 2y = 12 is (0, 6). State the x- and y-intercepts of these equations: 17. x 4y = 8 x-int: y-int: 18. 5x 4y = 20 x-int: y-int: 19. 6x + y = 12 x-int: y-int: 20. Draw a pair of parallel lines. 21. Draw a pair of perpendicular lines. 22. What is true about the slopes of parallel lines?. 23. One way of talking about the slopes of perpendicular lines is to say that the product of the slopes is -1. We could also say the slopes of perpendicular lines are. 24. Fill in the following grid: Original Line s Equation or Slope Original Slope Slope of a Parallel Line Slope of a Perpendicular Line Slope = 2/5 y = 6x + 1 y = 3 x y = 8 x = -2

4 Are the lines described below parallel to each other, perpendicular to each other, or neither? (HINT: Find each slope!) 25. The lines passing through (-4, 3) and (1,-3) and the line passing through (1, 2) and (-1, 3). 26. The line passing through (3, 9) and (-2, -1) and the graph of y = The line with x-intercept -2 and y-intercept 5 and the line with x-intercept 2 and y-intercept The line passing through (8, -4) and (4, 6) and the graph of 2x 5y = Write an equation of a line parallel to the line of y = 13x + 1:. 30. Write an equation of a line that is perpendicular to y = 13x + 1:. For problems 31 40, write an equation in Slope-Intercept Form for the line that satisfies each set of conditions. 31. Slope 1/3, passes through the origin 32. Slope 4, passes through (2, -1) 33. x-intercept 5, y-intercept Passes through (2, 4) and (-4, 7) 35. Horizontal and passes through (3, -1) 36. Vertical and passes through (-2, 5) 37. Passes through (6, -5), perpendicular to the line 38. Passes through (-3, -7), parallel to 1 2 whose equation is 3x y 7. (Hint: solve for y to find the slope) the graph of y x

5 For problems 31 40, write an equation in Slope-Intercept Form for the line described. 39. The line with these points: 40. The line based off this graph: x y Paul the plumber is coming in to fix a leak in your water pipe. He charges $40 to show up and $60 for each hour he works on your leak. (You could also say $100 for the first hour and $60 for each additional hour.) a) Define a pair of variables you could use for this situation: b) Write an equation in Slope Intercept Form that models this situation: c) How much would he charge you if Paul had 4 hours of work? d) If Paul charged you $160, how long was he there? For problems 42-51, graph the following equations on each coordinate plane y x y x y 2 x y 4 x 1

6 46. y x x 4y x y y x 51. y = - ½ x 5 3 Selected Answers (be sure to show your process for each question & that your answer matches the one below): 8a) (7, 3); m = 2 8b) (1, -4); m = -5 8c) (-4, 0); m = - 3 / 4 9.) y 2 = -1(x + 4) y = -x 2 10.) y + 4 = 3 / 2 (x + 2) y = 3 / 2 x 1 11.) y 2 = - 3 / 4 (x 2) y = - 3 / 4 x + 3½ 12.) y 12 = 5(x + 10) 13.) m = 1; b = ) m = -3; b = ) m = -7; b = 4 16.) m = 0; b = 9 y = 5x ) x: (8, 0); y: (0, -2) 18.) x: (4, 0); y: (0, -5) 19.)x:(2, 0); y:(0,12) 25.) Neither 26.) Neither 27.) Parallel 28.) Perpendicular 31.) y = 1 / 3 x 32.) y = 4x 9 33.) y = 6 / 5 x 6 34.) y = - 1 / 2 x ) y = ) x = ) y = 1 / 6 x 6 38.) y = 2 / 3 x 5 39.) y = -3x ) y = ½x + 1 Other answers (including graphs) can be discussed in class.

Chapter 2: Functions and Graphs Lesson Index & Summary

Chapter 2: Functions and Graphs Lesson Index & Summary Section 1: Relations and Graphs Cartesian coordinates Screen 2 Coordinate plane Screen 2 Domain of relation Screen 3 Graph of a relation Screen 3 Linear equation Screen 6 Ordered pairs Screen 1 Origin