Think Ahead, Tro Section 11.5 Chemistry 151 Linear & Exponential Functions Linear and Exponential Functions
You can get help with this work from the following sources:
1. Visit your instructor during office hours
2. Go to the Native American Center
3. Use your textbook: Appendix IB and ID (Pages A-4 toA-6).
Concept A: Know the general forms of linear vs. exponential equations & graphs
Linear functions have the general form: y = mx + b, where m is a constant* known as
the slope and b is a constant* known as the y-intercept. In contrast, exponential
functions have the general form: y = aebx, where a and b are constants* that can vary in
value and e (also known as exp) has a constant value of 2.718. Graphing a linear
function gives you a line; whereas graphing an exponential function, gives you a curve.
By carrying out tasks 1-3 below, you will gain a better sense for these two types of
functions.
*Strictly speaking these are “parameters” since they can vary from situation to situation.
1. Create a “linear” data set using the equation y = 3x + 2.
x
1
2
3
4
y
5
8
11
14
The first y value has been determined for you by multiplying 1 (the
value of x) by 3 and then adding 2. This equals 5—so that is the
first y value. Now you do the calculations to determine y when x =
2, 3 and 4.
2. Now create an “exponential” data set using the equation y = 3e2x.
x
1
2
3
4
y
22
164
1210
8943
Again, the first y-value has been determined for you. In this
case, e is raised to the power of 2—which is 1 (the value of x)
times 2; e2*1 is then multiplied by 3. Note to do this this with your
calculator, press the ex button (usually 2ndF ln), followed by 2;
enter; and then *3. This equals 22—so that is the first y value.
Now you do the calculations to determine y when x = 2, 3 and 4.
Think Ahead, Tro Section 11.5 Chemistry 151 Linear & Exponential Functions 3. Next, create graphs for each of these data sets using the grids provided below:
14 9000 12 8000 10 7000 8 6000 6 5000 4 4000 2 3000 0 1 2 3 4 2000 1000 0 1 2 3 4 Concept B: Know how to convert an exponential equation into a linear one.
Information is easily obtained from the slope or y-intercept of a linear function, so it is
often useful to be able to convert an exponential equation into a linear one. An example
of this in chemistry is when we convert the exponential relationship between vapor
pressure (Pvap) and temperature (T) into the Clausius-Clapeyron equation:
Pvap = β exp(-ΔHvap / RT)
into
ln Pvap = -ΔHvap / R * (1/T) + ln β
4. Determine the value of slope and y-intercept when the equation y = 3e2x is converted
from an exponential function into a linear one. Start by taking the natural logarithm
(ln) of both sides. Next, note that ln (a * b) = ln a + ln b; so this “splits” the 3 and ex.
Then note that ln (ex) = x; so this essentially “pulls” the exponent “out” of the
expression. Lastly, rearrange the equation into y = mx + b form to identify the slope
and y-intercept.
y = 3e2x → ln y = ln (3e2x) = ln 3 + ln (e2x) = 1.099 + 2x
ln y = 2x + 1.099
y
mx
b
Therefore the slope is 2 and the y-intercept is 1.099