12-4 Point-Slope Form

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpWrite the equation of the line that passes through each pair of points in slope-intercept form.

1. (0, –3) and (2, –3)

2. (5, –3) and (5, 1)

3. (–6, 0) and (0, –2)

4. (4, 6) and (–2, 0)

y = –3

x = 5

Course 3

12-4 Point-Slope Form

y = x + 2

y = – x – 213

Problem of the Day

Without using equations for horizontal or vertical lines, write the equations of four lines that form a square.

Possible answer: y = x + 2, y = x – 2, y = –x + 2, y = –x – 2

Course 3

12-4 Point-Slope Form

Learn to find the equation of a line given one point and the slope.

Course 3

12-4 Point-Slope Form

Vocabulary

point-slope form

Insert Lesson Title Here

Course 3

12-4 Point-Slope Form

Point on the line((xx11, , yy11))

Point-slope formyy – – yy11 = = mm ( (xx – – xx11))

slopeslope

The point-slope form of an equation of a line with slope m passing through (x1, y1) is y – y1 = m(x – x1).

Course 3

12-4 Point-Slope Form

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line.

y – 7 = 3(x – 4)

Additional Example 1A: Using Point-Slope Form to Identify Information About a Line

y – y1 = m(x – x1)

y – 7 = 3(x – 4)

m = 3

(x1, y1) = (4, 7)

The line defined by y – 7 = 3(x – 4) has slope 3, and passes through the point (4, 7).

The equation is in point-slope form. Read the value of m from the equation. Read the point from the equation.

Course 3

12-4 Point-Slope Form

y – 1 = (x + 6)

Additional Example 1B: Using Point-Slope Form to Identify Information About a Line

y – y1 = m(x – x1)

(x1, y1) = (–6, 1)

Rewrite using subtraction instead of addition.

13

13

y – 1 = (x + 6)

y – 1 = [x – (–6)]13

m =13

The line defined by y – 1 = (x + 6) has slope , and

passes through the point (–6, 1).

13

13

Course 3

12-4 Point-Slope Form

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line.

y – 5 = 2 (x – 2)

Check It Out: Example 1A

y – y1 = m(x – x1)

y – 5 = 2(x – 2)

m = 2

(x1, y1) = (2, 5)

The line defined by y – 5 = 2(x – 2) has slope 2, and passes through the point (2, 5).

The equation is in point-slope form. Read the value of m from the equation. Read the point from the equation.

Course 3

12-4 Point-Slope Form

y – 2 = (x + 3)

Check It Out: Example 1B

23

(x1, y1) = (–3, 2)

Rewrite using subtraction instead of addition.

23

y – 2 = (x + 3)

y – 2 = [x – (–3)]23

m =23

The line defined by y – 2 = (x + 3) has slope , and

passes through the point (–3, 2).

23

23

y – y1 = m(x – x1)

Course 3

12-4 Point-Slope Form

Write the point-slope form of the equation with the given slope that passes through the indicated point.

the line with slope 4 passing through (5, –2)

Additional Example 2A: Writing the Point-Slope Form of an Equation

y – y1 = m(x – x1)

The equation of the line with slope 4 that passes through (5, –2) in point-slope form is y + 2 = 4(x – 5).

Substitute 5 for x1, –2 for y1, and 4 for m.

[y – (–2)] = 4(x – 5)

y + 2 = 4(x – 5)

Course 3

12-4 Point-Slope Form

the line with slope –5 passing through (–3, 7)

Additional Example 2B: Writing the Point-Slope Form of an Equation

y – y1 = m(x – x1)

The equation of the line with slope –5 that passes through (–3, 7) in point-slope form is y – 7 = –5(x + 3).

Substitute –3 for x1, 7 for y1, and –5 for m.

y – 7 = -5[x – (–3)]

y – 7 = –5(x + 3)

Course 3

12-4 Point-Slope Form

Write the point-slope form of the equation with the given slope that passes through the indicated point.

the line with slope 2 passing through (2, –2)

Check It Out: Example 2A

y – y1 = m(x – x1)

The equation of the line with slope 2 that passes through (2, –2) in point-slope form is y + 2 = 2(x – 2).

Substitute 2 for x1, –2 for y1, and 2 for m.

[y – (–2)] = 2(x – 2)

y + 2 = 2(x – 2)

Course 3

12-4 Point-Slope Form

the line with slope –4 passing through (–2, 5)

Check It Out: Example 2B

y – y1 = m(x – x1)

The equation of the line with slope –4 that passes through (–2, 5) in point-slope form is y – 5 = –4(x + 2).

Substitute –2 for x1, 5 for y1, and –4 for m.

y – 5 = –4[x – (–2)]

y – 5 = –4(x + 2)

Course 3

12-4 Point-Slope Form

A roller coaster starts by ascending 20 feet for every 30 feet it moves forward. The coaster starts at a point 18 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 150 feet forward. Assume that the roller coaster travels in a straight line for the first 150 feet.

Additional Example 3: Entertainment Application

As x increases by 30, y increases by 20, so the slope

of the line is or . The line passes through the point (0, 18).

2030

23

Course 3

12-4 Point-Slope Form

Additional Example 3 Continued

y – y1 = m(x – x1) Substitute 0 for x1, 18 for y1,

and for m.23

The equation of the line the roller coaster travels along, in point-slope form, is y – 18 = x. Substitute 150 for x to find the value of y.

23

y – 18 = (150)23

y – 18 = 100

y – 18 = (x – 0)23

y = 118

The value of y is 118, so the roller coaster will be at a height of 118 feet after traveling 150 feet forward.

Course 3

12-4 Point-Slope Form

Check It Out: Example 3

A roller coaster starts by ascending 15 feet for every 45 feet it moves forward. The coaster starts at a point 15 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 300 feet forward. Assume that the roller coaster travels in a straight line for the first 300 feet.

As x increases by 45, y increases by 15, so the slope

of the line is or . The line passes through the point (0, 15).

1545

13

Course 3

12-4 Point-Slope Form

Check It Out: Example 3 Continued

y – y1 = m(x – x1) Substitute 0 for x1, 15 for y1,

and for m.13

The equation of the line the roller coaster travels along, in point-slope form, is y – 15 = x. Substitute 300 for x to find the value of y.

13

y – 15 = (300)13

y – 15 = 100

y – 15 = (x – 0)13

y = 115

The value of y is 115, so the roller coaster will be at a height of 115 feet after traveling 300 feet forward.

Course 3

12-4 Point-Slope Form

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line.

1. y + 6 = 2(x + 5)

2. y – 4 = – (x – 6)

Write the point-slope form of the equation with the given slope that passes through the indicated point.

3. the line with slope 4 passing through (3, 5)

4. the line with slope –2 passing through (–2, 4)

Lesson Quiz

(–5, –6), 2

Insert Lesson Title Here

y – 5 = 4(x – 3)

y – 4 = –2(x + 2)

25

(6, 4), – 25

Course 3

12-4 Point-Slope Form