Often, patterns can be used to describe and classify types of functions.
The data collected in the Hands-On
Algebra activity can be represented in a table to see a pattern more easily.
Each fold increases the number of sections by a factor of 2.
Let x equal the number of
folds and let y equal the number of sections. Then the function y =
2x represents the number of sections for any number of folds.
This is an example of an exponential
function. An exponential function is of the form y = ax,
where a > 0 and a ¹ 1. Some
people also refer to equations of the form y = abx + c
as exponential functions. For this form of the exponential function, the
value a is called the coefficient.
You can make a table to help graph
exponential functions.
1.
Graph y = 2x.
Alternative Solutions:
The graph of an exponential function
changes little for small x values. However, as the values of x increase,
the y values increase quickly.
The initial value of an exponential function is the
value of the function when x 0.
This is the same as the y-intercept. Exponential functions of the form y
= ax have an initial value of 1. Exponential functions of
the form y = abx + c have an initial value of c.
Example
Graph each exponential function. Then state the y-intercept.
2.
y = 5x
Alternative Solutions:
To
find the y-intercept, let x = 0 and solve for y.
y =
50 or 1
In
this case, the y-intercept is 1.
3.
y = 2x + 3
Alternative Solutions:
The y-intercept is 4. The
constant is 3: 1 + 3 = 4.
Quantities that increase rapidly have exponential growth.
For instance, money in the bank may grow at an exponential rate.
Finance
Link
4. When Taina was 10 years old, she received a certificate of deposit (CD)
for $2000 with an annual interest rate of 5%. After eight years, how much money
will she have in the account?
Alternative Solutions:
After
the first year, the CD is worth the initial deposit plus interest.
Taina
can make a table to look for patterns in the growth of her CD.
Let x represent
the number of years. Then the function that represents the balance in Taina’s
CD is B(x) = 2000(1.05)x. After eight years, it
will have a balance of $2954.91.
For exponential data, you can create a best-fit curve that
passes through most of the data points. You can then use this curve to make
predictions.
Sumber
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