Chapter Linear Equations and Functions Section.4 Writing the Equation of a Line Writing Equations of Lines Critical to a thorough understanding of linear equations and functions is the ability to write the equation of a line given different pieces of information. The following process will work for almost every situation you are presented with. Step 1: Determine the value of the slope, m. Step : Determine the coordinates of one ordered pair, (#, @ ). Step : Use the point-slope form,!@ @ = A(# # ), to find the equation of the line. Step 4: Rewrite the resulting equation in, slope-intercept form, y = mx + b. Step 5: When appropriate, rewrite the equation in function notation: f (x) = mx + b. Problem 14 MEDIA/CLASS EXAMPLE Writing Equations of Lines For each of the following, find the equation of the line that meets the following criteria: a) Slope m = 4 passing through the point (0, ). b) Passing through the points (0, ) and (1, 5) 95
Chapter Linear Equations and Functions c) Passing through the points (, ) and (4, 9) d) Parallel to y = x 7 and passing through (, 5) e) Horizontal line passing through (, 5). f) Vertical line passing through (, 5). 96
Chapter Linear Equations and Functions Parallel and Perpendicular Lines Two lines are parallel if they never intersect. It is clear that if two lines are parallel, then their slopes must be equal. The picture below will help us understand the relationship between the slopes of two lines that are perpendicular, that is, two lines that intersect at a right angle (90 g ). The slope of the bold line is A = e K a b a b The slope of the regular line is A 1 = K e (note that the sign of this slope is negative since this line is decreasing). We can also see that the two triangles are congruent, since they have sides of the same length. This means that their corresponding angles have the same measure as well. The sum of a bold angle with a regular angle is 90 g, since the third angle in each triangle is a right angle. Looking at the angles formed where the two lines meet, we can see the sum of one bold angle, with one regular angle, plus the angle between both lines adds up to 180 g. We can conclude that the angle between the lines is 90 g as well. In short, we have checked that: Slopes of Parallel and Perpendicular Lines Slopes of parallel lines are the EQUAL to each other. Slopes of perpendicular lines are NEGATIVE RECIPROCALS of each other. 97
Chapter Linear Equations and Functions Problem 15 WORKED EXAMPLE Writing Equations of Lines Write an equation of the line to satisfy each set of conditions. a) A line that contains the points (, 5) and (0, 1) Slope: Use the ordered pairs (, 5) and (0, 1) to compute slope. m = 1 5 4 4 = = 0 ( ) Ordered Pair: Choose one ordered pair: (0, 1). Point-Slope Form: Use the slope and the given point to write the equation of the line in point-slope form. Then solve for @ in order to re-write the equation on slopeintercept form. y y = m( x x1) Equation: Equation in slope-intercept form: Function notation: 1 4 y 1 = ( x 0) 4 y 1 = x 4 y = x + 1 4 f ( x) = x + 1 4 y = x + 1 b) A line that contains the points (7, ) and parallel to y = x + 6 Slope: The given line has slope. Since the two lines are parallel, both will have slope. Ordered Pair: Given point: (7, ) Point-Slope Form: Use the slope and the given point to write the equation of the line in point-slope form. Then solve for @ in order to re-write the equation on slopeintercept form. y y = m( x x1 ) y ( ) = ( x 7) y + = x 14 y = x 17 Equation: Equation in slope-intercept form: y = x 17 Function notation: f(x) = x 17 1 98
Chapter Linear Equations and Functions c) A line that contains the point ( 4, ) and perpendicular to x + y = 6 Slope: Rewrite the given line in slope-intercept form: The slope of given line is 1 6. x + y = 6 y = x + 6 y = x + The slope of the perpendicular line is the negative reciprocal of the given line s slope: 6 1 Ordered Pair: Given point: ( 4, ) Point-Slope Form: Use the slope and the given point to write the equation of the line in point-slope form. Then solve for @ in order to re-write the equation on slopeintercept form. y = m( x x ) y 1 1 y ( ) = ( x ( 4)) y + = ( x + 4) y + = x + 6 y = x + Equation: Equation in slope-intercept form: Function Notation: f ( x) = x + y = x + 99
Chapter Linear Equations and Functions Problem 16 YOU TRY Writing Equations of Lines a) Find the equation of the line passing through the points (1, 4) and (, ) and write your answer in the form f (x) = mx + b. Show complete work in this space. b) What is the y-intercept for this equation? Show work or explain your result. c) What is the x-intercept for this equation? Show complete work to find this. 100
Chapter Linear Equations and Functions Problem 17 WORKED EXAMPLE Writing Linear Equations from Graphs A line has the following graph: Slope: Identify two ordered pairs from the graph and use them to determine the slope. (5, 0) and (0, ) m = (0) 0 (5) = 5 = 5 Ordered Pair: Find an ordered pair that is on the line: (5, 0) Point-Slope Form: Use the slope and the given point to write the equation of the line in point-slope form. Then solve for @ in order to re-write the equation on slopeintercept form. y y = m( x x ) 1 1 y 0 = ( x 5) 5 y = x 5 Equation: Equation in slope-intercept form: @ = 6 S # Rewrite in function notation: " # = 6 S # 101
Chapter Linear Equations and Functions Problem 18 YOU TRY Writing Linear Equations from Graphs Use the given graph of the function f below to help answer the questions below. Assume the line intersects grid corners at integer (not decimal) values. a) Is the line above increasing, decreasing, or constant? b) What is the y-intercept? Also, plot and label the y-intercept on the graph. c) What is the x-intercept? Also, plot and label the x-intercept on the graph. d) What is the slope? Show your work. e) Write the equation of the line using function notation. 10
Chapter Linear Equations and Functions Problem 19 MEDIA/CLASS EXAMPLE Applications of Linear Functions A candy company has a machine that produces candy canes. The number of candy canes, C, produced depends on the amount of time, t, the machine has been operating. The machine produces 160 candy canes in five minutes. In twenty minutes, the machine can produce 640 candy canes. a)! Determine a linear equation to model this situation. Write your equation in function notation. b)! Determine the vertical intercept of this linear equation. Write it as an ordered pair and interpret its practical meaning. c)! Determine the horizontal intercept of this linear equation. Write it as an ordered pair and interpret its practical meaning. d)! How many candy canes will this machine produce in 1 minute? e)! How many candy canes will this machine produce in 1 hour? 10
Chapter Linear Equations and Functions Problem 0 YOU TRY Applications of Linear Functions The graph below shows a person s distance from home as a function of time. a)! Identify the vertical intercept. Write it as an ordered pair and interpret its practical meaning. b)! Identify the horizontal intercept. Write it as an ordered pair and interpret its practical meaning. c)! Determine a linear equation to model this situation where t represents time and D represents the distance from home. Write your equation in function notation. d)! How far has this person traveled in one minute? 104