2 4 Writing Linear Functions Warm Up Lesson

2 -4 Writing Linear Functions Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 2 2

2 -4 Writing Linear Functions Warm Up Write each function in slope-intercept form. 1. 4 x + y = 8 y = – 4 x + 8 2. –y = 3 x y = – 3 x 3. 2 y = 10 – 6 x y = – 3 x + 5 Determine whether each line is vertical or horizontal. 3 4. x = 4 vertical Holt Mc. Dougal Algebra 2 5. y = 0 horizontal

2 -4 Writing Linear Functions Objectives Use slope-intercept form and pointslope form to write linear functions. Write linear functions to solve problems. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Vocabulary Point-slope form Holt Mc. Dougal Algebra 2

2 -4 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 Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 1: Writing the Slope-Intercept Form of the Equation of a Line Write the equation of the graphed line in slopeintercept form. Step 1 Identify the y-intercept. The y-intercept b is 1. Holt Mc. Dougal Algebra 2

2 -4 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. 3 4 – 3 Slope is rise = – 3. run Holt Mc. Dougal Algebra 2 4 4

2 -4 Writing Linear Functions Example 1 Continued Step 3 Write the equation in slope-intercept form. y = mx + b 3 y=– x+1 4 m= – 3 and b = 1. 4 The equation of the line is y = – Holt Mc. Dougal Algebra 2 3 x + 1. 4

2 -4 Writing Linear Functions Check It Out! Example 1 Write the equation of the graphed line in slopeintercept form. Step 1 Identify the y-intercept. The y-intercept b is 3. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Check It Out! Example 1 Continued 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 Slope is rise = 3. run Holt Mc. Dougal Algebra 2 4 3 3 4

2 -4 Writing Linear Functions Check It Out! Example 1 Continued Step 3 Write the equation in slope-intercept form. y = mx + b y= 3 x+3 4 m= 3 and b = 3. 4 The equation of the line is y = Holt Mc. Dougal Algebra 2 3 x + 3. 4

2 -4 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 Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Helpful Hint If you reverse the order of the points in Example 2 B, the slope is still the same. m= 6 – 16 5 – 11 Holt Mc. Dougal Algebra 2 = – 10 – 6 = 5 3

2 -4 Writing Linear Functions Example 2 A: 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 Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 2 B: Finding the Slope of a Line Given Two or More Points Find the slope of the line. x y 4 2 8 5 12 8 16 11 Choose any Let (x 1, y 1) be (4, 2) and (x 2, y 2) be (8, 5). two points. Use the slope formula. The slope of the line is 3. 4 Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 2 C: Finding the Slope of a Line Given Two or More Points 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. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Check It Out! Example 2 A Find the slope of the line. Let (x 1, y 1) be (– 4, – 1) and (x 2, y 2) be (– 2, 1). x y – 6 – 3 – 4 – 1 – 2 1 Choose any two points. Use the slope formula. The slope of the line is 1. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Check It Out! Example 2 B Find the slope of the line through (2, – 5) and (– 3, – 5). Let (x 1, y 1) be (2, – 5) and (x 2, y 2) be (– 3, – 5). Use the slope formula. The slope of the line is 0. Holt Mc. Dougal Algebra 2

2 -4 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 Mc. Dougal Algebra 2

2 -4 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. x – 8 – 4 y – 5 – 3. 5 4 – 0. 5 8 1 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 Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 3 Continued Method A Point-Slope Form Using (8, 1): Rewrite in slopeintercept form. y – y 1 = m(x – x 1) Substitute. Simplify. Holt Mc. Dougal Algebra 2 Distribute. Solve for y.

2 -4 Writing Linear Functions Example 3 Continued Method B Slope-intercept Form Using (8, 1), solve for b. y = mx + b Rewrite the equation using m and b. y = mx + b Substitute. 1=3+b Simplify. b = – 2 Solve for b. The equation of the line is Holt Mc. Dougal Algebra 2 .

2 -4 Writing Linear Functions Check It Out! Example 3 a Write the equation of the line in slope-intercept form with slope – 5 through (1, 3). Method A Point-Slope Form y – y 1 = m(x – x 1) y – (3) = – 5(x – 1) y – 3 = – 5(x – 1) Substitute. Simplify. Rewrite in slope-intercept form. y – 3 = – 5(x – 1) y – 3 = – 5 x + 5 y = – 5 x + 8 Holt Mc. Dougal Algebra 2 Distribute. Solve for y. The equation of the slope is y = – 5 x + 8.

2 -4 Writing Linear Functions Check It Out! Example 3 b Write the equation of the line in slope-intercept form through (– 2, – 3) and (2, 5). First, find the slope. Let (x 1, y 1) be (– 2, – 3) and (x 2, y 2) be (2, 5). Method B Slope-Intercept Form y = mx + b Rewrite the equation 5 = (2)2 + b using m and b. 5=4+b y = mx + b y = 2 x + 1 1=b The equation of the line is y = 2 x + 1. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 4 A: Entertainment Application The table shows the rents and selling prices of properties from a game. Selling Price Rent ($) Express the rent as a function of the selling price. Let x = selling price and y = rent. Find the slope by choosing two points. Let (x 1, y 1) be (75, 9) and (x 2, y 2) be (90, 12). Holt Mc. Dougal Algebra 2 75 9 90 12 160 26 250 44

2 -4 Writing Linear Functions Example 4 A Continued To find the equation for the rent function, use point-slope form. y – y 1 = m(x – x 1) Use the data in the first row of the table. Simplify. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 4 B: Entertainment Application Graph the relationship between the selling price and the rent. How much is the rent for a property with a selling price of $230? To find the rent for a property, use the graph or substitute its selling price of $230 into the function. Substitute. y = 46 – 6 y = 40 The rent for the property is $40. Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Check It Out! Example 4 a Express the cost as a linear function of the number of items. Let x = items and y = cost. Find the slope by choosing two points. Let (x 1, y 1) be (4, 14) and (x 2, y 2) be (7, 21. 50). Holt Mc. Dougal Algebra 2 Items Cost ($) 4 14. 00 7 21. 50 18

2 -4 Writing Linear Functions Check It Out! Example 4 a Continued To find the equation for the number of items, use point-slope form. y – y 1 = m(x – x 1) y – 14 = 2. 5(x – 4) y = 2. 5 x + 4 Holt Mc. Dougal Algebra 2 Use the data in the first row of the table. Simplify.

2 -4 Writing Linear Functions Check It Out! Example 4 b Graph the relationship between the number of items and the cost. Find the cost of 18 items. To find the cost, use the graph or substitute the number of items into the function. y = 2. 5(18) + 4 Substitute. y = 45 + 4 y = 49 The cost for 18 items is $49. Holt Mc. Dougal Algebra 2

2 -4 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 Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Holt Mc. Dougal Algebra 2

2 -4 Writing Linear Functions Example 5 A: Writing Equations of Parallel and Perpendicular Lines Write the equation of the line in slope-intercept form. parallel to y = 1. 8 x + 3 and through (5, 2) m = 1. 8 Parallel lines have equal slopes. Use y – y 1 = m(x – x 1) with (x 1, y 1) = y – 2 = 1. 8(x – 5) (5, 2). y – 2 = 1. 8 x – 9 y = 1. 8 x – 7 Holt Mc. Dougal Algebra 2 Distributive property. Simplify.

2 -4 Writing Linear Functions Example 5 B: Writing Equations of Parallel and Perpendicular Lines 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, Use y – y 1 = m(x – x 1). y + 2 is equivalent to y – (– 2). Distributive property. Simplify. Holt Mc. Dougal Algebra 2 .

2 -4 Writing Linear Functions Check It Out! Example 5 a Write the equation of the line in slope-intercept form. parallel to y = 5 x – 3 and through (1, 4) m=5 Parallel lines have equal slopes. y – 4 = 5(x – 1) Use y – y 1 = m(x – x 1) with (x 1, y 1) = (5, 2). y – 4 = 5 x – 5 Distributive property. y = 5 x – 1 Holt Mc. Dougal Algebra 2 Simplify.

2 -4 Writing Linear Functions Example 5 B: Writing Equations of Parallel and Perpendicular Lines 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, Use y – y 1 = m(x – x 1). y + 2 is equivalent to y – (– 2). Distributive property. Simplify. Holt Mc. Dougal Algebra 2 .

2 -4 Writing Linear Functions Check It Out! Example 5 b Write the equation of the line in slope-intercept form. perpendicular to and through (0, – 2) The slope of the given line is , so the slope of the perpendicular, line is the opposite reciprocal Use y – y 1 = m(x – x 1). y + 2 is equivalent to y – (– 2). Distributive property. Simplify. Holt Mc. Dougal Algebra 2 .

2 -4 Writing Linear Functions Lesson Quiz: Part I Write the equation of each line in slopeintercept form. 1. y = – 2 x – 1 2. parallel to y = 0. 5 x + 2 and through (6, 1) y = 0. 5 x – 2 3. perpendicular to Holt Mc. Dougal Algebra 2 and through (4, 4)

2 -4 Writing Linear Functions Lesson Quiz: Part II 4. Express the catering cost as a function of the number of people. Find the cost of catering a meal for 24 people. Number in Group Cost ($) 4 7 15 98 134 230 f(x) = 12 x + 50; $338 Holt Mc. Dougal Algebra 2