Transcript Slide 1

Slope and Direct Variation
m  74
m0
m   125
m is undefined
12548
11580
m  968

 2316fish/year
2000 1995
5
Slope and Direct Variation
A direct variation is described by an equation of the
form
y = kx, where k ≠ 0
We say that y varies directly with x or y varies
directly as x.
In the equation, y = kx, k is the constant of variation.
Slope and Constant of Variation
Name the constant of
variation for the equation.
The constant of
Then find the slope of the
variation is 2.
line that passes through the
pair of points.
Slope formula
Simplify.
Answer: k = 2 and m = 2.
Notice that the slope of the graph of y = kx is k.
For a direct variation, the slope and constant of variation are
identical.
Slope and Constant of Variation
Name the constant of
variation for the equation.
Then find the slope of the
line that passes through the
pair of points.
The constant of
variation is – 4.
Slope formula
Simplify.
Answer: k = – 4 and m = – 4.
The ordered pair (0, 0) is a solution of y = kx.
Therefore, the graph of y = kx passes through the origin. You can
use this information to graph direct variation equations.
Slope and Constant of Variation
Name the constant of variation for the equation.
Then find the slope of the line that passes through
the pair of points.
a.
Answer: constant of
variation: 4;
slope: 4
Slope and Constant of Variation
Name the constant of variation for the equation.
Then find the slope of the line that passes through
the pair of points.
b.
Answer: constant of
variation: –3;
slope: –3
The ordered pair (0, 0) is a solution of y = kx.
Therefore, the graph of y = kx passes through the origin. You can
use this information to graph direct variation equations.
Direct Variation with k > 0
Step 1
Write the slope as
a ratio.
Step 2
Graph (0, 0).
Step 3
From the point (0, 0), move up 1 unit and right
1 unit. Draw a dot.
Step 4
Draw a line containing the points.
Direct Variation with k > 0
Answer:
Direct Variation with k < 0
Step 1
Write the slope as
a ratio.
Step 2
Graph (0, 0).
Step 3
From the point (0, 0), move down 3 units and
right 2 units. Draw a dot.
Step 4
Draw a line containing the points.
Direct Variation with k < 0
Answer:
Slope and Direct Variation
k is the constant of variation
Write and Solve a Direct Variation Equation
Suppose y varies directly as x, and
when
Write a direct variation equation that relates x and y.
Find the value of k.
Direct variation formula
Replace y with 9 and x with –3.
Divide each side by –3.
Simplify.
Answer: Therefore,
Write and Solve a Direct Variation Equation
Use the direct variation equation to find x when
Direct variation equation
Replace y with 15.
Divide each side by –3.
Simplify.
Answer: Therefore,
when
Write and Solve a Direct Variation Equation
Suppose y varies directly as x, and
when
a. Write a direct variation equation that relates x and y.
Answer:
b. Use the direct variation equation to find x when
Answer: x = –15
Write this in your notes:
y
k
x
Direct Variation Equation
Travel The Ramirez family is driving cross-country
on vacation. They drive 330 miles in 5.5 hours.
Write a direct variation equation to find the distance
driven for any number of hours.
Words
The distance traveled is 330 miles, and the
time is 5.5 hours.
Variables
Distance
Equation
330 mi
equals
rate
r
times
time.
5.5h
Direct Variation Equation
Solve for the rate.
Original equation
Divide each side by 5.5.
Simplify.
Answer: Therefore, the direct variation equation is
Write this in your notes:
d  rt
Direct Variation Equation
Graph the equation.
The graph of
slope of 60.
Answer:
passes through the origin with a
Direct Variation Equation
Estimate how many hours it would take to drive
600 miles.
Original equation
Replace d with 600.
Divide each side by 60.
Simplify.
Answer: At this rate, it will take 10 hours to drive
600 miles.