Lesson 2-3 Objective The student will be able to: 1) write equations using slope-intercept form. 2) identify slope and y-intercept from an equation

Important!!! This is one of the big concepts in algebra. You need to thoroughly understand this! Slope – Intercept Form y = mx + b m represents the slope b represents the y-intercept

Writing Equations When asked to write an equation, you need to know two things – slope (m) and y-intercept (b). There are three types of problems you will face.

Writing Equations – Type #1 Write an equation in slope-intercept form of the line that has a slope of 2 and a y-intercept of 6. To write an equation, you need two things: slope (m) = y – intercept (b) = We have both!! Plug them into slope-intercept form y = mx + b y = 2x

Write the equation of a line that has a y-intercept of -3 and a slope of y = -3x – 4 2.y = -4x – 3 3.y = -3x y = -4x + 3

Writing Equations – Type #2 Write an equation of the line that has a slope of 3 and goes through the point (2,1). To write an equation, you need two things: slope (m) = y – intercept (b) = We have to find the y-intercept!! Plug in the slope and ordered pair into y = mx + b 1 = 3(2) + b 3 ???

Writing Equations – Type #2 1 = 3(2) + b Solve the equation for b 1 = 6 + b = b To write an equation, you need two things: slope (m) = y – intercept (b) = y = 3x

Writing Equations – Type #3 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! First, we have to find the slope. Plug the points into the slope formula. Simplify ???

Writing Equations – Type #3 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (4, 2) because both numbers are positive. 2 = (4) + b ???

Writing Equations – Type #3 2 = (4) + b Solve the equation for b 2 = + b To write an equation, you need two things: slope (m) = y – intercept (b) =

Write an equation of the line that goes through the points (0, 1) and (1, 4). 1.y = 3x y = 3x y = -3x y = -3x + 1

To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b. Find the slope and y-intercept. 1)y = 3x – 7 y = mx + b m = 3, b = -7

Find the slope and y-intercept. 2)y = x y = mx + b y = x + 0 3)y = 5 y = mx + b y = 0x + 5 m = b = 0 m = 0 b = 5

-3 Find the slope and y-intercept. 4)5x - 3y = 6 Write it in slope-intercept form. (y = mx + b) 5x – 3y = 6 -3y = -5x + 6 y = x - 2 m = b = -2

Write it in slope-intercept form. (y = mx + b) 2y + 2 = 4x 2y = 4x - 2 y = 2x - 1 Find the slope and y-intercept. 5) 2y + 2 = 4x 222 m = 2 b = -1

Find the slope and y-intercept of y = -2x m = 2; b = 4 2.m = 4; b = 2 3.m = -2; b = 4 4.m = 4; b = -2

Graphing a Line Given a Point & Slope Graph a line though the point (2, -6) with m=2/3 Graph (2, -6) Count up 2 for the rise, and to the right 3 for the run Plot the point, repeat, then connect x y

Graphing Lines m = - ½ b = 3 Plot y-intercept (b) Use the slope to find two more points Connect x y

Graph the Line (3, -3) m = undefined x y

POINT-SLOPE FORM Chapter 2-3

WarPPPPPpm-Up BELLWORK – Thurs Write an equation of the line in slope-intercept form. 2. passes through (–2, 2) and (1, 8) ANSWER 1. passes through (3, 4), m = 3 y = 2x + 6 y = 3x – 5

Warm-Up 3. A carnival charges an entrance fee and a ticket fee. One person paid $27.50 and brought 5 tickets. Another paid $45.00 and brought 12 tickets. How much will 22 tickets cost? ANSWER $70

Example 1 Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2. y – y 1 = m(x – x 1 ) Write point-slope form. y + 3 = 2(x – 4) Substitute 2 for m, 4 for x 1, and –3 for y 1.

Guided Practice Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2. 1. y – 4 = –2(x + 1) ANSWER

Example 2 y + 2 = (x – 3). 2 3 Graph the equation SOLUTION Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2). 2 3 Plot the point ( 3, –2 ). Find a second point on the line using the slope. Draw a line through both points.

Guided Practice – Graph the equation 2. ANSWER y – 1 = (x – 2).

Example 3 Write an equation in point- slope form of the line shown. SOLUTION STEP 1 = y 2 – y 1 x 2 – x 1 m = 3 – 1 –1 – 1 = 2 –2 = –1 Find the slope of the line.

Example 3 Method 1 Method 2 Use (–1, 3). Use (1, 1). y – y 1 = m(x – x 1 ) y – 3 = –(x +1)y – 1 = –(x – 1) STEP 2 Write the equation in point-slope form. You can use either given point. CHECK Check that the equations are equivalent by writing them in slope-intercept form. y – 3 = –x – 1 y = –x + 2 y – 1 = –x + 1 y = –x + 2

Guided Practice Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). 3. y – 3 = (x – 2) or 1 2 y – 4 = (x – 4) 1 2 ANSWER

Example 5 WORKING RANCH The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group.

Lesson Quiz ANSWER y + 4 = –2(x – 6) Write an equation in point-slope form of the line that passes through (6, –4) and has slope Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10). 2. ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

Lesson Quiz A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7 -day trip. 3. ANSWER C = 235t + 170, where C is total cost and t is time (in days); $1815

Write Equations and Parallel and Perpendicular Lines

Warm-Up Are the lines parallel? Explain. 2. –x = y + 4, 3x + 3y = 5 ANSWER 1.y – 2 = 2x, 2x + y = 7 Yes; both slopes are –1. No; one slope is 2 and the other is –2.

Example 1 SOLUTION Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1. STEP 1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

Example 1 STEP 2 Find the y- intercept. Use the slope and the given point. y = mx + by = mx + b –5 = 3(–3) + b 4 = b Write slope-intercept form. Substitute 3 for m,  3 for x, and  5 for y. Solve for b. STEP 3 Write an equation. Use y = mx + b. y = 3x + 4 Substitute 3 for m and 4 for b.

Guided Practice 1. Write an equation of the line that passes through (–2, 11 ) and is parallel to the line y = –x + 5. y = –x + 9 ANSWER

Example 2 Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b: x + 5y = 2 Line c: –10y – 2x = 0 SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form. The slope is 5. Write the equations for lines b and c in slope-intercept form.

Example 2 Line b: x + 5y = 2 5y = – x + 2 Line c: –10y – 2x = 0 –10y = 2x y = – x 1 5 x – ANSWER Lines b and c have slopes of –, so they are parallel. Line a has a slope of 5, the negative reciprocal of –, so it is perpendicular to lines b and c

Guided Practice Determine which lines, if any, are parallel or perpendicular. Line a: 2x + 6y = –3 Line b: y = 3x – 8 Line c: –1.5y + 4.5x = 6 ANSWER parallel: b and c ; perpendicular: a and b, a and c

Example 3 SOLUTION Line a: 12y = –7x + 42 Line b: 11y = 16x – 52 Find the slopes of the lines. Write the equations in slope-intercept form. The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they ? STATE FLAG

Example 3 Line a: 12y = –7x + 42 Line b: 11y = 16x – 52 y = –y = – x y = x – ANSWER The slope of line a is –. The slope of line b is. The two slopes are not negative reciprocals, so lines a and b are not perpendicular

Guided Practice 3. Is line a perpendicular to line b? Justify your answer using slopes. Line a: 2y + x = –12 Line b: 2y = 3x – 8 ANSWER No; the slope of line a is –, the slope of line b is. The slopes are not negative reciprocals so the lines are not perpendicular

Example 4 SOLUTION Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3. STEP 1 Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is. 1 2 –

Example 4 STEP 2 Find the y- intercept. Use the slope and the given point. Write slope-intercept form. –5 = – (4) + b 1 2 Substitute – for m, 4 for x, and – 5 for y. 1 2 y =y = mx + bmx + b –3 = b Solve for b. STEP 3 Write an equation. y = mx + b Write slope-intercept form. y = – x – Substitute – for m and –3 for b. 1 2

Guided Practice 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7. y = – x ANSWER

Lesson Quiz 1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2. y = 5x + 9 ANSWER Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x – 2. y = 4x + 3 ANSWER

Lesson Quiz 3. Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes. ANSWER Paths a and b are perpendicular because their slopes, 2 and are negative reciprocals. No paths are parallel. 1 2 –

Graphing Lines Graph the line perpendicular to y = 2x + 3 that goes through the point (-2, 3). The slope of the line is 2 so the slope of the perpendicular line is -1/2. m = -1/2 b = 3

Graphing Lines Graph the line parallel to x = -1 that goes through the point (3, -3). The slope of the line is undefined (VUX) so the slope of the parallel line is undefined.

Writing Equations in Standard Form MFCR Lesson 2-3

Warm-Up ANSWER 1.(1, 4), (6, –1) y + 2 = 3(x + 1) or y – 7 = 3(x – 2) y – 4 = –(x – 1) or y + 1 = –(x – 6) 2.( –1, –2), (2, 7) Write an equation in point-slope form of the line that passes through the given points.

Example 1 To write another equivalent equation, multiply each side by x – 12y = 8 To write one equivalent equation, multiply each side by 2. SOLUTION Write two equations in standard form that are equivalent to 2x – 6y = 4. x – 3y = 2

Example 2 SOLUTION y – y 1 = m(x – x 1 ) Calculate the slope. STEP 1 –3–3 m =m = 1 – (–2) 1 – 2 = 3 –1 = Write an equation in point-slope form. Use (1, 1). Write point-slope form. y – 1 = –3(x – 1) Substitute 1 for y 1,  3 for m and 1 for x 1. Write an equation in standard form of the line shown. STEP 2

Example 2 Rewrite the equation in standard form. 3x + y = 4 Simplify. Collect variable terms on one side, constants on the other. STEP 3

Guided Practice Write an equation in standard form of the line through (3, –1) and (2, –3). 2. –2x + y = –7 ANSWER

Example 3 SOLUTION Write an equation of the specified line. The y- coordinate of the given point on the blue line is –4. This means that all points on the line have a y- coordinate of –4. An equation of the line is y = –4. a.a. The x- coordinate of the given point on the red line is 4. This means that all points on the line have an x- coordinate of 4. An equation of the line is x = 4. b.b. Blue line a.a. Red line b.b.

Guided Practice Write equations of the horizontal and vertical lines that pass through the given point. 3. (–8, –9) y = –9, x = –8 ANSWER 4. (13, –5) y = –5, x = 13 ANSWER

Lesson Quiz Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points. ANSWER 2x + y = –4 1. (1, –6), m = –2 2. (–4, –3), (2, 9) ANSWER –2x + y = 5