Download Unit 2.5 - Slope

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Period #: _______ Date: ____________________
Notes……Unit 2.5 Slope
Objective:
To make a connection between direct variation and slope.
Resources
Direct variation – A
proportional relationship that
creates a continuous straight
line that goes through the
y
origin and where
is a
x
constant. The equation is
y = kx.
Slope – The measure of
steepness of a line (rise over
run)
A direct variation equation is y = kx. The “k” in this equation is called the
constant of proportionality or the constant of variation. It is found by doing
y
the calculation . We can also recall that the rate of change is a ratio that
x
compares the change in the output (y) to the change in the input (x). Which is
y
the same as . Another term, slope, is the measure of the steepness of a line
x
y
(rise (y) over run (x)) which is also . So, these terms are used
x
interchangeably.
But, not all functions (or equations) are direct variation functions (y = kx).
Remember the famous slope-intercept equation of a line, y = mx + b? The “m”
in the equation is the same as the “k” in the direct variation function. But the
“b” is the y-intercept in this equation and since the direct variation function has
to go through the origin (0, 0), then the y-intercept is 0.
x-intercept – where the
graph crosses the x-axis.
y-intercept – where the
graph crosses the y-axis.
To find the rate of change in a linear function that has the form y = mx + b, we
still need to divide y by x, but in a different way. We call it the “change of y
y
divided by the change of x” which looks like:
.
x
Example 1: In a table, we can find rate of change like this:
Table
y-intercept
Equation of the line
x
y
0
3
2
6
4
9
6
12
8
15
10
18
Resources
Example 2: In a graph, we can find rate of change like this:
Graph
y-intercept
Equation of the line
Slope Formula – the
change of y over the change
of x.
m
y2  y1
x2  x1
Point-slope formula –
y – y1 = m(x – x1)
Example 3: When given two points, we can find the rate of change like this:
Given (2, 4) and (3, 10)
Slope-intercept form –
First find the slope using the slope formula:
y = mx + b
m
y2  y1
=
x2  x1
Second, choose one of these options to find the equation of the line:
Function Notation – is
essentially replacing "y = "
in your equations with "f(x)
=".
Point-slope formula:
y – y1 = m(x – x1)
OR
slope-intercept form:
y = mx + b
Example: Change y =2x +
1 to f(x) = 2x + 1
Example 4: f(x) = 2x + 6 when x = 2, 20, 200, -2
f(2) = ______________
f(20) = _____________
Assignment: Unit 2.5 Classwork (All problems)
f(200) = ______________
f(-2) = __________