Slope intercept

5-6 Slope-Intercept Form Holt Algebra 1 Lesson Quiz Lesson Presentation Warm Up

Warm Up Find each y -intercept. 1. y = 3 x + 2 2. 5 x – 3 y = 12 Find each slope. 3. 5. 4 x + 2y = 10 6. 3 x + 2 = 6 y Solve each equation for y. 4. 6 x + 2 y = 6 2 – 4 – 3 y = –2 x + 5

Write a linear equation in slope-intercept form. Graph a line using slope-intercept form. Objectives

You have seen that you can graph a line if you know two points on the line. Another way is to use the point that contains the y- intercept and the slope of the line.

Example 1A: Graphing by Using Slope and y- intercept Graph the line given the slope and y- intercept. y intercept = 4 Step 1 The y -intercept is 4 , so the line contains (0, 4 ). Plot (0, 4). Step 3 Draw the line through the two points. • • • • Run = 5 Rise = –2 y Step 2 Slope = Count 2 units down and 5 units right from (0, 4) and plot another point.

Step 3 Draw the line through the two points. • • Rise = 4 Example 1B: Graphing by Using Slope and y- intercept Graph the line given the slope and y- intercept. slope = 4; y -intercept = Step 1 The y -intercept is , so the line contains (0, ). Plot (0, ). Run = 1 Step 2 Slope = Count 4 units up and 1 unit right from (0, ) and plot another point.

Check It Out! Example 1a Graph the line given the slope and y- intercept. slope = 2, y -intercept = –3 Step 1 The y -intercept is –3, so the line contains (0, –3). Plot (0, –3). • • Step 3 Draw the line through the two points. Rise = 2 Run = 1 Step 2 Slope = Count 2 units up and 1 unit right from (0, –3) and plot another point.

Graph each line given the slope and y- intercept. slope = , y -intercept = 1 Check It Out! Example 1b Step 1 The y -intercept is 1, so the line contains (0, 1). Plot (0, 1). • Step 3 Draw the line through the two points. • Rise = –2 Run = 3 Step 2 Slope = Count 2 units down and 3 units right from (0, 1) and plot another point.

If you know the slope of a line and the y -intercept, you can write an equation that describes the line. Step 1 If a line has a slope of 2 and the y- intercept is 3 , then m = 2 and (0, 3) is on the line. Substitute these values into the slope formula.

Step 2 Solve for y : Simplify the denominator. 2 x + 3 = y , or y = 2 x + 3 Multiply both sides by x. Add 3 to both sides. • • 2 x = y – 3 + 3 +3

Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y -intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

Example 2A: Writing linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form. y = m x + b Substitute the given values for m and b. Simply if necessary. slope = ; y -intercept = 4 y = x + 4

slope = –9; y -intercept = y = m x + b Substitute the given values for m and b. Simply if necessary. Example 2B: Writing linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form. y = –9 x +

slope = 3; y -intercept = y = m x + b Substitute the given values for m and b. Simply if necessary. Example 2C: Writing linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form.

slope = ; y -intercept = –6 y = m x + b Substitute the given values for m and b. Simply if necessary. Example 2D: Writing linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form.

slope = 2; (3, 4) is on the line y = m x + b Substitute 2 for m, 3 for x, and 4 for y. 4 = 2 (3) + b Step 1 Find the y -intercept. Write the slope-intercept form. Solve for b. Since 6 is added to b, subtract 6 from both sides to undo the addition. Example 2E: Writing linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form. – 2 = b 4 = 6 + b – 6 –6

Example 2E Continued Step 2 Write the equation. y = m x + b Write the slope-intercept form. Substitute 2 for m, and –2 for b. y = 2 x + (–2) y = 2 x – 2 slope = 2; (3, 4) is on the line Write the equation that describes the line in slope-intercept form.

Check It Out! Example 2 A line has a slope of 8 and (3, –1) is on the line. Write the equation that describes this line in slope-intercept form. Step 1 Find the y -intercept. y = m x + b Substitute 8 for m, 3 for x, and –1 for y. – 1 = 8 (3) + b Write the slope-intercept form. Solve for b. Since 24 is added to b, subtract 24 from both sides to undo the addition. – 25 = b – 1 = 24 + b – 24 –24

Check It Out! Example 2 Continued Step 2 Write the equation. y = m x + b Write the slope-intercept form. Substitute 8 for m, and –25 for b. y = 8 x + (–25) y = 8 x – 25 A line has a slope of 8 and (3, –1) is on the line. Write the equation that describes this line in slope-intercept form.

Example 3A: Using Slope-Intercept Form to Graph Write the equation in slope-intercept form. Then graph the line described by the equation. y = 3 x – 1 Step 1 Plot (0, –1 ). Step 2 Count 3 units up and 1 unit right and plot another point. Step 3 Draw the line connecting the two points. • • y = 3 x – 1 is in the form y = m x + b slope: m = 3 = y- intercept: b = –1

Example 3B: Using Slope-Intercept Form to Graph Write the equation in slope-intercept form. Then graph the line described by the equation. 2 y + 3 x = 6 Step 1 Write the equation in slope-intercept form by solving for y. Subtract 3x from both sides. Since y is multiplied by 2, divide both sides by 2. 2 y + 3 x = 6 – 3 x –3 x 2 y = –3 x + 6

Example 3B Continued Step 2 Graph the line. Plot (0, 3 ). • Count 3 units down and 2 units right and plot another point. • Draw the line connecting the two points. • • Write the equation in slope-intercept form. Then graph the line described by the equation. slope: m = y -intercept: b = 3 is in the form y = m x + b .

Check It Out! Example 3a Write the equation in slope-intercept form. Then graph the line described by the equation. is in the form y = m x + b .

Check It Out! Example 3a Continued Step 2 Graph the line. y -intercept: b = 0 Step 1 Plot (0, 0 ). Step 2 Count 2 units up and 3 units right and plot another point. Step 3 Draw the line connecting the two points. • • Write the equation in slope-intercept form. Then graph the line described by the equation. y = x + 0 is in the form y = m x + b . slope:

Check It Out! Example 3b 6 x + 2 y = 10 Step 1 Write the equation in slope intercept form by solving for y. Subtract 6x from both sides. Since y is multiplied by 2, divide both sides by 2. Write the equation in slope-intercept form. Then graph the line described by the equation. 6 x + 2 y = 10 – 6 x –6 x 2 y = –6 x + 10

Check It Out! Example 3b Continued Step 2 Graph the line. y = –3 x + 5 is in the form y = m x + b . y -intercept: b = 0 • Plot (0, 5 ). • • Count 3 units down and 1 unit right and plot another point. • Draw the line connecting the two points. • Write the equation in slope-intercept form. Then graph the line described by the equation. slope: m =

Check It Out! Example 3c y = –4 Step 1 Plot (0, –4 ). y = –4 is in the form y = m x + b . y- intercept: b = –4 slope: m = 0 = = 0 Since the slope is 0, the line will be a horizontal at y = –4. • Write the equation in slope-intercept form. Then graph the line described by the equation.

Example 4: Application A closet organizer charges a $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.

Example 4: Application a. Write an equation that represents the cost as a function of the number of hours. An equation is y = 30 x + 100. A closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below. Cost is $30 for each hour plus $100 y = 30 • x + 100

Example 4 Continued A closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below. b. Identify the slope and y -intercept and describe their meanings. The y -intercept is 100. This is the cost for 0 hours, or the initial fee of $100. The slope is 30. This is the rate of change of the cost: $30 per hour. c. Find the cost if the organizer works 12 hrs. y = 30 x + 100 = 30 (12) + 100 = 460 Substitute 12 for x in the equation The cost of the organizer for 12 hours is $460.

Check It Out! Example 4 A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph. a. Write an equation that represents the cost as a function of the number of guests. An equation is y = 18 x + 200. Cost is $18 for each meal plus $200 y = 18 • x + 200

Check It Out! Example 4 Continued A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph. b. Identify the slope and y -intercept and describe their meanings. The y -intercept is 200. This is the cost for 0 people, or the initial fee of $200. The slope is 18. This is the rate of change of the cost: $18 per person. c. Find the cost of catering an event for 200 guests. y = 18 x + 200 = 18 (200) + 200 = 3800 Substitute 200 for x in the equation The cost of catering for 200 people is $3800.

Lesson Quiz: Part I Write the equation that describes each line in the slope-intercept form. 1. slope = 3, y -intercept = –2 y = 3 x – 2 2. slope = 0, y -intercept = 3. slope = , (2, 7) is on the line y = y = x + 4

Lesson Quiz: Part II Write each equation in slope-intercept form. Then graph the line described by the equation. 4. 6 x + 2 y = 10 y = –3 x + 5 5. x – y = 6 y = x – 6

Slope intercept

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Slope intercept