Holt Algebra Writing Linear Functions Recall from Lesson 2-3 that the slope-intercept form of a linear equation is y= mx + b, where m is the slope of the line and b is its y-intercept. In Lesson 2-3, you graphed lines when you were given the slope and y-intercept. In this lesson you will write linear functions when you are given graphs of lines or problems that can be modeled with a linear function. Holt Algebra Writing Linear Functions Example 1: Writing the Slope-Intercept Form of the Equation of a Line Write the equation of the graphed line in slope- intercept form. Step 1 Identify the y-intercept. The y-intercept b is 1. Holt Algebra Writing Linear Functions Example 1 Continued Step 2 Find the slope. Choose any two convenient points on the line, such as (0, 1) and (4, 2). Count from (0, 1) to (4, 2) to find the rise and the run. The rise is 3 units and the run is 4 units. Slope is = = . rise run 3 Holt Algebra Writing Linear Functions Example 1 Continued Step 3 Write the equation in slope-intercept form. y = mx + b 3434 y = x + 1 m = and b = The equation of the line is 3434 y = x + 1. Holt Algebra Writing Linear Functions Write the equation of the graphed line in slope- intercept form. Step 1 Identify the y-intercept. The y-intercept b is 3. Check It Out! Example 1 Holt Algebra Writing Linear Functions Step 2 Find the slope. Choose any two convenient points on the line, such as (4, 0) and (0, 3). Count from (4, 0) to (0, 3) to find the rise and the run. The rise is 3 units and the run is 4 units Check It Out! Example 1 Continued 3 Slope is =. rise run Holt Algebra Writing Linear Functions Step 3 Write the equation in slope-intercept form. Check It Out! Example 1 Continued y = mx + b 3434 y = x + 3 m = and b = The equation of the line is 3434 y = x + 3. Holt Algebra Writing Linear Functions Notice that for two points on a line, the rise is the differences in the y-coordinates, and the run is the differences in the x-coordinates. Using this information, we can define the slope of a line by using a formula. Holt Algebra Writing Linear Functions If you reverse the order of the points in Example 2B, the slope is still the same. m = = Helpful Hint 6 16 5 11 10 6 5 3 Holt Algebra Writing Linear Functions Example 2A: Finding the Slope of a Line Given Two or More Points Find the slope of the line through (1, 1) and (2, 5). Let (x 1, y 1 ) be (1, 1) and (x 2, y 2 ) be (2, 5). Use the slope formula. The slope of the line is 2. Holt Algebra Writing Linear Functions Example 2B: Finding the Slope of a Line Given Two or More Points Find the slope of the line. x y Let (x 1, y 1 ) be (4, 2) and (x 2, y 2 ) be (8, 5). Choose any two points. Use the slope formula. The slope of the line is. 3434 Holt Algebra Writing Linear Functions Find the slope of the line shown. Let (x 1, y 1 ) be (0,2) and (x 2, y 2 ) be (1, 2). The slope of the line is 0. Example 2C: Finding the Slope of a Line Given Two or More Points Holt Algebra Writing Linear Functions Find the slope of the line. x664422 y311 Let (x 1, y 1 ) be (4, 1) and (x 2, y 2 ) be (2, 1). Choose any two points. Use the slope formula. The slope of the line is 1. Check It Out! Example 2A Holt Algebra Writing Linear Functions Let (x 1, y 1 ) be (2, 5) and (x 2, y 2 ) be (3, 5). The slope of the line is 0. Check It Out! Example 2B Find the slope of the line through (2,5) and (3, 5). Use the slope formula. Holt Algebra Writing Linear Functions Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form. Holt Algebra Writing Linear Functions Example 3: Writing Equations of Lines In slope-intercept form, write the equation of the line that contains the points in the table. x84 48 y53.50.51 First, find the slope. Let (x 1, y 1 ) be (8, 5) and (x 2, y 2 ) be (8, 1). Next, choose a point, and use either form of the equation of a line. Holt Algebra Writing Linear Functions Example 3 Continued Method A Point-Slope Form Rewrite in slope- intercept form. Substitute. Simplify. Solve for y. Distribute. Using (8, 1): y y 1 = m(x x 1 ) Holt Algebra Writing Linear Functions Method B Slope-intercept Form Substitute. Simplify. Solve for b. Rewrite the equation using m and b. Using (8, 1), solve for b. y = mx + b b = 2 y = mx + b 1 = 3 + b The equation of the line is. Example 3 Continued Holt Algebra Writing Linear Functions Method A Point-Slope Form Rewrite in slope-intercept form. Substitute. Simplify. Solve for y. Distribute. Write the equation of the line in slope-intercept form with slope 5 through (1, 3). Check It Out! Example 3a The equation of the slope is y = 5x + 8. y y 1 = m(x x 1 ) y (3) = 5(x 1) y 3 = 5(x 1) y 3 = 5x + 5 y = 5x + 8 Holt Algebra Writing Linear Functions By comparing slopes, you can determine if the lines are parallel or perpendicular. You can also write equations of lines that meet certain criteria. Holt Algebra Writing Linear Functions Holt Algebra Writing Linear Functions Parallel Lines have the same slopes Holt Algebra Writing Linear Functions Example 5A: Writing Equations of Parallel and Perpendicular Lines Parallel lines have equal slopes. Use y y 1 = m(x x 1 ) with (x 1, y 1 ) = (5, 2). Distributive property. Simplify. m = 1.8 y 2 = 1.8(x 5) y 2 = 1.8x 9 y = 1.8x 7 Write the equation of the line in slope-intercept form. parallel to y = 1.8x + 3 and through (5, 2) Holt Algebra Writing Linear Functions Perpendicular Lines have the opposite reciprocal slopes Example 5: Line 1 and Line 2 are perpendicular. Find the slope of Line 2. a.Line 1 has a slope Line 2 has a slope: b.Line 1 has a slope -6 Line 2 has a slope: Holt Algebra Writing Linear Functions Example 5B: Writing Equations of Parallel and Perpendicular Lines Distributive property. Simplify. Use y y 1 = m(x x 1 ). y + 2 is equivalent to y (2). Write the equation of the line in slope-intercept form. perpendicular to and through (9, 2) The slope of the given line is, so the slope of the perpendicular line is the opposite reciprocal,. Holt Algebra Writing Linear Functions Write the equation of the line in slope-intercept form. parallel to y = 5x 3 and through (1, 4) Parallel lines have equal slopes. Use y y 1 = m(x x 1 ) with (x 1, y 1 ) = (5, 2). Distributive property. Simplify. m = 5 y 4 = 5(x 1) y 4 = 5x 5 y = 5x 1 Check It Out! Example 5a Holt Algebra Writing Linear Functions Example 5B: Writing Equations of Parallel and Perpendicular Lines Distributive property. Simplify. Use y y 1 = m(x x 1 ). y + 2 is equivalent to y (2). Write the equation of the line in slope-intercept form. perpendicular to and through (9, 2) The slope of the given line is, so the slope of the perpendicular line is the opposite reciprocal,. Holt Algebra Writing Linear Functions Distributive property. Simplify. Use y y 1 = m(x x 1 ). y + 2 is equivalent to y (2). Check It Out! Example 5b The slope of the given line is, so the slope of the perpendicular, line is the opposite reciprocal. Write the equation of the line in slope-intercept form. perpendicular to and through (0, 2) Holt Algebra Writing Linear Functions Lesson Quiz: Part I y = 2x 1 y = 0.5x 2 Write the equation of each line in slope- intercept form parallel to y = 0.5x + 2 and through (6, 1) 3. perpendicular to and through (4, 4) Holt Algebra Writing Linear Functions Lesson Quiz: Part II Number in GroupCost ($) Express the catering cost as a function of the number of people. Find the cost of catering a meal for 24 people. f(x) = 12x + 50; $338