Question

For each table of values, select whether the table represents a linear or exponential relationship. Please help 10 points for the answer

Answer 1

We have a Table here called Function A. So we need to know whether the table represents a linear or exponential relationship. Remember that:

A linear function has the form:

$f(x)=mx+b$ so if $x$ increases by a constant amount the functions value will have a common difference.

An exponential function has the form:

$f(x)=ab^x$ so if $x$ increases by a constant amount the functions value will have a common difference.

FUNCTION A)

EXPONENTIAL RELATIONSHIP.

Notice that in this case the function increases by a constant amount and in this case this amount is 3 because:

$6-3=3 \\ \\ 9-6=3 \\ \\ 12-9=3$

As you can see, it is true that $x$ increase by a constant amount of 3. Hence, we need to analyze the function values.

First, let's see if the function values has a common difference:

$9.252-6.083=3.169 \\ \\ 14.07-9.252=4.818 \\ \\ Stop \ here!$

We have stopped because there is no any common difference. Let's try to see if there is common ratio:

$\frac{9.252}{6.083}=1.52 \\ \\ \frac{14.07}{9.252}=1.52 \\ \\ \frac{21.4}{14.07}=1.52$

As you can see these function values have a common ratio. In conclusion THIS TABLE REPRESENTS AN EXPONENTIAL RELATIONSHIP.

FUNCTION B)

LINEAR RELATIONSHIP.

From the Table, it's easy to realize that this is a linear relationship by taking a look on the function values. Notice that $x$ increases here at a constant amount of 2 because:

$6-4=2 \\ \\ 8-6=2 \\ \\ 10-8=2$

So, let's see if the function values has a common difference:

$14-7=7 \\ \\ 21-14=7 \\ \\ 28-21=7$

As you can see these function values have a common difference. In conclusion THIS TABLE REPRESENTS A LINEAR RELATIONSHIP.

FUNCTION C)

EXPONENTIAL RELATIONSHIP.

It is likely that this is an exponential function, just take a look at the function values. Notice that $x$ increases here at a constant amount of 2 because:

$2-0=2 \\ \\ 4-2=2 \\ \\ 6-2=2$

So, let's see if the function values has a common ratio:

$\frac{4}{16}=0.25 \\ \\ \frac{1}{4}=0.25 \\ \\ \frac{0.25}{1}=0.25$

As you can see these function values have a common ratio. In conclusion THIS TABLE REPRESENTS AN EXPONENTIAL RELATIONSHIP.