Understanding the difference Linear equations These equations take the form y mx b m is the slope of the line b is the value of y when x 0 the y ID: 255438
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Slide1
Linear vs. Exponential Graphing
Understanding the differenceSlide2
Linear equations
These equations take the form:
y = mx
+
b
m
is the slope of the line
b
is the value of
y
when
x
= 0
(the
y
-
intercept
)Slide3
Graphing linear equations
For the equation:
y = 5x - 2 we can insert
any value for x
and determine the
corresponding y
value.
The infinite number of solutions that satisfies this equation forms a line with a slope of 5 that intersects the y-axis at (0, -2)
Let’s graph…Slide4
Here we chose the
x-values indicated below
, determined the corresponding y-values, and plotted.
Note, however, these are not the only solutions…..
Continue…Slide5
Note that this line actually extends
infinitely, in both directions!Slide6
Exponential equations
These equations express an exponential relationship. Let’s consider a common problem--dandelions!Slide7
The dandelion problem
Each dandelion forms a puffball with several hundred seeds that can be scattered by the wind. Each of these seeds has the potential to form a new plant…
The extent to which your lawn will become infested with dandelions will vary, but let us look at an imaginary situation…Slide8
Joseph Greenlawn has a beautiful yard. One day he spots a single pretty yellow flower and he decides to let it grow. That dandelion eventually goes to seed and produces a new crop later in the year. Joseph doesn’t appreciate the extent of the problem, and goes away on a extended trip for 16 months to study the vast number of grasses worldwide.
What does he find when he returns?
?
Continue…Slide9
Let’s make some assumptions (I’m not a gardener, so these are NOT facts!!)
Each dandelion produces 100 seeds.
Of these 100 seeds, only 4 will land in Joseph’s lawn and make a new plant.
Joseph lives in a temperate climate where a new crop of dandelions emerges every 4 months.
Continue…Slide10
How many dandelions will Joseph find upon his return?
Time (months)
# Dandelions
0 1
4 4
8 16
12 64
16 256
Uh oh!!!Slide11
This is an exponential relationship…
With every increase in x (here, time in months), y (the number of dandelions) increases exponentially.
If dandelion growth were linear, the increase in y (number of dandelions) would be constant with a constant increase in x (time in months) because this is the SLOPE.
Let’s compare linear and exponential growth…Slide12
How many dandelions would Joseph find if dandelion growth were linear?
Time (months)
# Dandelions
0 1
4 4
8 7
12 10
16 13
This assumes that the increase seen from zero to 4 months remains constant.Slide13
Let’s see how these two situations look graphically…
Note two things for linear vs. exponential growth:
First:
Linear growth is, of course, shown by a line
Exponential growth is a
steeply rising curve
Second:
Observe how much greater the number of dandelions is at 15 months during exponential vs. linear growth!!