lesson 2-3 objective the student will be able to: 1) write equations using slope-intercept form. 2)...
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Lesson 2-3 ObjectiveThe 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 #1Write 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 + 6
2
6
Write the equation of a line that has a y-intercept of -3 and a slope of -4.
1. y = -3x – 4
2. y = -4x – 3
3. y = -3x + 4
4. y = -4x + 3
Writing Equations – Type #2Write 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 #21 = 3(2) + b
Solve the equation for b
1 = 6 + b
-6 -6
-5 = b
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
y = 3x - 5
3
-5
Writing Equations – Type #3Write 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.
Simplify2 1
4 ( 2)m
??????
1
6m
Writing Equations – Type #3Write 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
1
6???
1
6
Writing Equations – Type #3
2 = (4) + bSolve the equation for b
2 = + b
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
1
62
32
3
11
3b
2
3
1
6 11
3
1 11
6 3y x
Write an equation of the line that goes through the points (0, 1) and (1, 4).
1. y = 3x + 4
2. y = 3x + 1
3. y = -3x + 4
4. 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
2
3m =
b = 0
2
3
2
3
m = 0b = 5
-3 -3 -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 - 25
3m =
b = -2
5
3
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
2 2 2
m = 2b = -1
Find the slope and y-intercept of y = -2x + 4
1. 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-UpBELLWORK – Thurs. 10-24-13
Write an equation of the line in slope-intercept form.
2. passes through (–2, 2) and (1, 8)
ANSWER
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 – y1 = m(x – x1) Write point-slope form.
y + 3 = 2(x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.
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 secondpoint on the line using the slope.Draw a line through both points.
Guided Practice
–Graph the equation2.
ANSWER
y – 1 = (x – 2).
Example 3
Write an equation in point-slope form of the line shown.
SOLUTION
STEP 1
=y2 – y1
x2 – x1
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 – y1 = m(x – x1) y – y1 = m(x – x1)
y – 3 = –(x +1) y – 1 = –(x – 1)
STEP 2Write 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 – 1y = –x + 2
y – 1 = –x + 1y = –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) or1
2y – 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 2.
1.
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
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 2Find the y-intercept. Use the slope and the given point.
y = 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 + 9ANSWER
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 = – x15xy = 2
515 +–
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.
15
15
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 = – x + 1242 7
12
1152
y = x –1611
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.
712
1611
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.
12
32
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 thegiven point.
Write slope-intercept form.
–5 = – (4) + b12
Substitute – for m, 4 for x, and
–5 for y.
12
y = mx + b
–3 = b Solve for b.
STEP 3 Write an equation.
y = mx + b Write slope-intercept form.
y = – x – 312 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 + 414ANSWER
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 + 2.1
4–
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
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 0.5.
4x – 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 – y1 = m(x – x1)
Calculate the slope.STEP 1
–3m =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 y1, 3 for m
and 1 for x1.
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 = –7ANSWER
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.
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.
Blue linea. Red lineb.
Guided Practice
Write equations of the horizontal and vertical lines that pass through the given point.
3. (–8, –9)
y = –9, x = –8ANSWER
4. (13, –5)
y = –5, x = 13ANSWER
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
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