Download 5.3 Write Linear Equations in Point

5.3 Write Linear Equations in
Point-Slope Form
Point-Slope Form
y – y1 = m(x – x1)
m = slope and it passes through (x1, y1)
Y-intercept is not clear in this form unless you
solve it for y.
Example
• Using Point-Slope Form write an equation for
a line with a slope of ½ that goes through
(7, -2).
EXAMPLE 1
Write an equation in point-slope form
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.
EXAMPLE
1
Example
1
Write an for
equation
in point-slope
form
GUIDED PRACTICE
1. Write an equation in point-slope form of the line that passes
through the point (–1, 4) and has a slope of –2.
ANSWER
y – 4 = –2(x + 1)
Graphing an Equation in Point-Slope Form
• y – y1 = m(x – x1) it goes through (x1, y1)
• To graph:
Method 1:
Change the signs on x1 and y1 then plot the point.
Use the slope to find the next point.
Method 2
• Solve for y and use slope-intercept form
y – y1 = m(x – x1)
Distribute m, add y1.
EXAMPLE 2
Graph an equation in point-slope form
Graph the equation
y + 2 = 2 (x – 3).
3
SOLUTION
Because the equation is in point-slope form, you know that
the line has a slope of 2/3 and passes through the point
(3, –2).
Plot the point (3, –2). Find a second
point on the line using the slope.
Draw a line through both points.
EXAMPLE
2
Example
2
Graph anfor
equation
in point-slope
form
GUIDED PRACTICE
2.
Graph the equation
ANSWER
y – 1 = – (x – 2).
Example
• Graph y + 4 = 3(x – 1)
• Graph y – 1 = ½(x + 2)
Writing an Equation in Point-Slope
Form Given Two Points
• Find the slope
• Plug either point into the formula
• There are two possible answers, but they are
identical if solved for y.
• Example: Write an equation in point-slope
form that goes through the points (-1, 3) and
(1, 1)
Extra Example
• Write an equation in point slope form that
goes through (-2, 3) and (1, -3)
EXAMPLE
3
for Example
Use point-slope
form to3 write an equation
GUIDED PRACTICE
3. Write an equation in point-slope form of the line that passes
through the points (2, 3) and (4, 4).
ANSWER
1
1
y – 3 = (x – 2) or y – 4 = (x – 4)
2
2
EXAMPLE 4
Solve a multi-step problem
STICKERS
You are designing a sticker to advertise your band. A
company charges $225 for the first 1000 stickers and $80 for
each additional 1000 stickers. Write an equation that gives
the total cost (in dollars) of stickers as a function of the
number (in thousands) of stickers ordered. Find the cost of
9000 stickers.
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Identify the rate of change and a data pair. Let C be the cost
(in dollars) and s be the number of stickers (in thousands).
Rate of change, m: $80 per 1 thousand stickers
Data pair (s1, C1): (1 thousand stickers, $225)
EXAMPLE 4
Solve a multi-step problem
STEP 2
Write an equation using point-slope form. Rewrite the
equation in slope-intercept form so that cost is a function
of the number of stickers.
C – C1 = m(s – s1)
C – 225 = 80(s – 1)
C = 80s + 145
Write point-slope form.
Substitute 80 for m, 1 for s1, and 225 for C1.
Solve for C.
EXAMPLE 4
Solve a multi-step problem
STEP 3
Find the cost of 9000 stickers.
C = 80(9) + 145 = 865
Substitute 9 for s. Simplify.
ANSWER
The cost of 9000 stickers is $865.
Example
• A radio station charges $650 for the first
minute of ad time and then $340 for each
additional minute. Write an equation that
gives the total cost (in dollars) to run an ad as
a function of the number of minutes the ad
runs. Find the cost of 7 minutes of ad time.
Is it Linear
• To tell if a situation can be modeled or
graphed by a linear equation, the slope or rate
of change must be constant.
EXAMPLE 5
Write a real-world linear model from a table
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.
Number of people
4
6
8
10
12
Cost (dollars)
250
350
450
550
650
EXAMPLE 5
Write a real-world linear model from a table
SOLUTION
STEP 1
Find the rate of change for consecutive data pairs in the table.
350 – 250
550 – 450
450 – 350
650 – 550
= 50,
= 50,
= 50,
= 50
6–4
10 – 8
8–6
12 – 10
Because the cost increases at a constant rate of $50 per
person, the situation can be modeled by a linear equation.
EXAMPLE 5
Write a real-world linear model from a table
STEP 2
Use point-slope form to write the equation. Let C be the
cost (in dollars) and p be the number of people. Use the
data pair (4, 250).
C – C1 = m(p – p1)
C – 250 = 50(p – 4)
C = 50p +50
Write point-slope form.
Substitute 50 for m, 4 for p1, and 250 for C1.
Solve for C.
Example
• The table shows the
Time (hrs)
cost of renting a canoe
for different times (in
2
hours). Can the
situation be modeled by
4
a linear equation?
Explain. If possible,
6
write an equation that
gives the cost as a
8
function of time (in
10
hours).
Cost ($)
22
32
42
52
62
Guided Practice
• Top of page 305
Homework
• 1, 2 – 12 ev, 14 – 19, 20 – 34 ev, 38 – 42ev
• Bonus: 35, 36, 44