Try It! Write Exponential Functions
3. Write an exponential function for each set of points.
a. , and

Ask by Savage Dunn. in the United States

Jan 22,2025

Question

Try It! Write Exponential Functions 3. Write an exponential function for each set of points. a. $$ (1,12),(2,48),(3,192) $$, and $$ (4,768) $$

UpStudy AI · Accepted Answer

To find an exponential function that fits the given set of points $$(1, 12)$$, $$(2, 48)$$, $$(3, 192)$$, and $$(4, 768)$$, we'll follow these steps:

### Step 1: Understand the Form of an Exponential Function

An exponential function typically has the form:

$$ f(x) = a \cdot b^x $$

where:
- $$ a $$ is the initial value (the value of the function when $$ x = 0 $$),
- $$ b $$ is the base of the exponential function (the factor by which the function grows or decays).

### Step 2: Set Up Equations Using the Given Points

We can substitute each point into the general form to create a system of equations. Let's use the first two points to find $$ a $$ and $$ b $$.

1. For the point $$(1, 12)$$:
$$ 12 = a \cdot b^1 $$
$$ 12 = a \cdot b \quad \text{(Equation 1)} $$

2. For the point $$(2, 48)$$:
$$ 48 = a \cdot b^2 $$
$$ 48 = a \cdot b^2 \quad \text{(Equation 2)} $$

### Step 3: Solve for $$ b $$

Divide Equation 2 by Equation 1 to eliminate $$ a $$:

$$
\frac{48}{12} = \frac{a \cdot b^2}{a \cdot b} \
4 = b
$$

So, $$ b = 4 $$.

### Step 4: Solve for $$ a $$

Now that we have $$ b $$, substitute it back into Equation 1 to find $$ a $$:

$$
12 = a \cdot 4 \
a = \frac{12}{4} \
a = 3
$$

### Step 5: Write the Exponential Function

Now that we have both $$ a $$ and $$ b $$, we can write the exponential function:

$$
f(x) = 3 \cdot 4^x
$$

### Verification

Let's verify that this function fits all the given points:

1. **For $$ x = 1 $$:**
$$
f(1) = 3 \cdot 4^1 = 3 \cdot 4 = 12
$$
Matches $$(1, 12)$$.

2. **For $$ x = 2 $$:**
$$
f(2) = 3 \cdot 4^2 = 3 \cdot 16 = 48
$$
Matches $$(2, 48)$$.

3. **For $$ x = 3 $$:**
$$
f(3) = 3 \cdot 4^3 = 3 \cdot 64 = 192
$$
Matches $$(3, 192)$$.

4. **For $$ x = 4 $$:**
$$
f(4) = 3 \cdot 4^4 = 3 \cdot 256 = 768
$$
Matches $$(4, 768)$$.

### Final Answer

The exponential function that fits the given set of points is:

$$
f(x) = 3 \cdot 4^x
$$
The exponential function is $$ f(x) = 3 \cdot 4^x $$.

Try It! Write Exponential Functions
3. Write an exponential function for each set of points.
a. , and

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