GRAPHS OF FUNCTIONS
Mr. Thauvette
DP SL Mathematics
Review of Equations of Lines
Objective: Using the Point-Slope Form to Write an Equation
Recall that when the slope of a line and a point on the line are
known, the equation of the line can also be found. To do this, we
use the slope formula to write the slope of a line that passes
through points (x, y) and (x1, y1).
y - y1
m=
x - x1
We multiply both sides of this equation by
y - y1 = m ( x - x1 )
x - x1 to obtain
This form is called the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line
Example 1
Write an equation of the line with
slope –3 and containing the point
(1, –5). Write the equation in
slope-intercept form, y = mx + b
Example 1 – Solution
Because we know the slope and a point on the line, we use the
Point-slope form with m = –3 and (x1, y1) = (1, -5).
In slope-intercept form, the equation is y = – 3x – 2
Practice 1
Write an equation of the line with
slope –2 and containing the point
(2, –4). Write the equation in
slope-intercept form, y = mx + b
Answer: y = –2x
Example 2
Write an equation of the line through points (4, 0) and (–4, –5).
Write the equation in standard form Ax + By = C.
Solution: First we find the slope of the line.
Next we make use of the point-slope form. We replace (x1, y1) by
either (4, 0) or (–4, –5) in the point-slope form. We will choose the
Point (4, 0). The line through (4, 0) with slope 5/8 is as follows.
Continued on next slide
Example 2 – Solution
Let’s multiply through by 8 so that the coefficients are integers and
are less tedious to work with.
Continued on next slide
Example 2 – Solution
If we multiply both sides of –5x + 8y = –20 by –1, we have an
Equivalent equation in standard form. Both –5x + 8y = –20 and
5x – 8y = 20 are acceptable.
Practice 2
Write an equation of the line
through points (3, 0) and (–2, 4).
Write the equation in standard
form, Ax + By = C.
Answer: 4x + 5y = 12
Objective: Writing Equations of
Vertical and Horizontal Lines
A few special types of linear equations are those whose graphs are
vertical and horizontal lines.
Example 3.
Solution: Recall that a horizontal line has an equation of the form
y = c. Since the line contains the point (2, 3), the equation is y = 3
Practice 3
Write an equation of the
horizontal line containing the point
(–1, 6).
Answer: y = 6
Example 4
Write an equation of the line
containing the point
(2, 3) with undefined slope.
Solution: Since the line has
undefined slope, the line must be
vertical. A vertical line has an
equation of the form x = c, and
since the line contains the point
(2, 3), the equation is x = 2.
Practice 4
Write an equation of the line
containing the point (4, 7) with
Undefined slope.
Answer: x = 4
Objective: Writing Equations of
Parallel and Perpendicular Lines
Recall: Nonvertical parallel lines have the same slope and nonvertical
perpendicular lines have slopes whose product is –1.
Example 5
Solution: Because the line we want to find is parallel to the line
2x + y = –6, the two lines must have equal slopes. So we first find the
Slope of 2x + y = –6 by solving the equation for y to write it in the
Form y = mx + b. Here y = –2x – 6, so the slope is –2.
Example 5
Now we use the point-slope form to write an equation of the line
through (4, 4) with slope –2.
Continued on next slide
Example 5
The equation, y = –2x – 6, and the new equation, y = –2x + 12
Have the same slope but different y-intercepts so their graphs are
parallel, as shown. Also,
the graph of y = –2x + 12
contains the point (4, 4),
as desired.
Practice 5
Write an equation of the line containing the point (–1, 2) and
parallel to the line 3x + y = 5. Write the equation in
slope-intercept form y = mx + b
Answer: y = –3x – 1
Example 6
Write an equation of the line containing the point (–2, 1) and
perpendicular to the line 3x + 5y = 4. Write the equation in
slope-intercept form, y = mx + b.
Solution: First we find the slope of 3x + 5y = 4 by solving the
equation for y.
Continued on next slide
Example 6
Continued on next slide
Example 6
Continued on next slide
Example 6
The equation
and the new equation
Have negative reciprocal slopes,
So their graphs are perpendicular.
Also, the graph of
contains the point (–2, 1), as
desired.
Practice 6
Write an equation of the line containing the point (3, 4) and
perpendicular to the line 2x + 4y = 5. Write the equation in
slope-intercept form.
Answer: y = 2x – 2
Objective: Using the Point-Slope Form
in Applications
Example 7.
Southern Star Realty is an establish real estate company that has
enjoyed constant growth in sales since 2001. In 2005 the company
sold 250 houses, and in 2009 the company sold 330 houses. Use
these figures to predict the number of houses this company will
sell in 2013
Continued on next slide
Example 7
Solution:
1. UNDERSTAND. Read and reread the problem. Then let
x = the number of years after 2001
y = the number of houses sold in the year corresponding to x
The information provided then gives the ordered pairs (4, 250) and
(8, 330). To better visualize the sales of Southern Star Realty, we
graph the line that passes through the points (4, 250) and (8, 330).
Continued on next slide
Example 7
Continued on next slide
Example 7
2. TRANSLATE. We write an equation of the line that passes through
the points (4, 250) and (8, 330). To do so, we first find the slope of
the line.
Then, using the point-slope form to write the equation, we have
Continued on next slide
Example 7
3. SOLVE. To predict the number of houses sold in 2013, we use
y = 20x + 170 and complete the ordered pair (12, ), since
2013 – 2001 = 12.
4. INTERPRET.
CHECK: Verify that the point (12, 410) is a point on the line graphed
in step 1.
STATE: Southern Star Realty should expect to sell 410 houses in 2013.
Practice 7
Southern Regional is an established office product maintenance
company that has enjoyed constant growth in new maintenance
contracts since 2001. In 2003, the company obtained 15 new
contracts, and in 2009, the company obtained 33 new contracts. Use
these figures to predict the number of new contracts this company
can expect in 2017.
Answer: 57 new contracts
Practice 8
A rock is dropped from the top of a 400-foot building. After 1
Second, the rock is traveling 32 feet per second. After 3 seconds, the
Rock is traveling 96 feet per second. Let y be the rate of descent
And x be the number of seconds since the rock was dropped.
(a) Write a linear equation that relates time x to rate y.
(b) Use this equation to determine the rate of travel of the rock
4 seconds after it was dropped.
Answer: (a) y = 32x; (b) 128 feet per second
Practice 9
The Whammo Company has learned that by pricing a newly
released Frisbee at $6, sales will reach 2000 per day. Raising the
price to $8 will cause the sales to fall to 1500 per day. Assume that
the ratio of change in price to change in daily sales is constant, and
let x be the price of the Frisbee and y be number of sales.
(a) Find the linear equation that models the price-sales relationship
for this Frisbee.
(b) Use this equation to predict the daily sales of Frisbees if the price
is set at $7.50
Answer: (a) y = –250x + 3500; (b) 1625 Frisbees
Practice 10
In 2009, the average price of a new home sold in the United States
Was $267,900. In 2004, the average price of a new home in the
United States was $271,500. Let y be the average price of a new
Home in the year x, where x = 0 represents the year 2004.
(a) Write a linear equation that models the average price of a new
home in terms in the year x.
(b) Use this equation to predict the average price of a new home
in 2012.
Answer: (a) y = –720x + 271,500; (b) $265,740
Practice 11
The number of people employed in the United States as registered
Nurses was 2619 thousand in 2008. By 2018, this number is
Expected to rise to 3200 thousand. Let y be the number of registered
Nurses (in thousands) employed in the United States in the year x,
Where x = 0 represents 2008.
(a) Write a linear equation that models the number of people
(in thousands) employed as registered nurses in year x.
(b) Use this equation to estimate the number of people who will be
employed as registered in 2012.
Answer: (a) y = 58.1x + 2619; (b) 2851.4 thousand
Finding equations of lines from graphs
Find an equation of each line graphed in standard form.
Answer: 2x + y = 3
Answer: 2x – 3y = –7
Summary
Concept Extension – Practice 12
The graph shows the cost function, C(x), and the revenue function, R(x),
for a transistor manufacturer, where x is the number of units sold.
Concept Extension – Practice 13
A Water Supply company decides to change its rates to increase the
user-pays component of water supply and it phases in the procedure.
The new method included a fixed charge (possibly different for each
property depending on its assessed value) and a rate of 45 cents per
kilolitre for the first part used and 65 cents per kilolitre for the second
part.
In a particular bill, the charge for a quarter is $150 plus a charge for
up to 50kl at the lower rate and the rest at the higher rate. If the water
used is 63kl what will be the total bill?
45 cents per kilolitre for the first part used and 65 cents per kilolitre for the second part.
Practice 13
In a particular bill, the charge for a quarter is $150 plus a charge for up to 50kl at the lower rate
and the rest at the higher rate. If the water used is 63kl what will be the total bill?
Let the volume of water used be denoted by v kL, then, based on the information given, we
have that
1.
for v £ 50, the total cost is given by 150 + 0.45v
2.
for v > 50, the total cost is given by 172.5 + 0.65(v – 50)
72
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