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Write an exponential function that includes the following points:

(1, 6) and (2, 12)

Step 1 - Solve for [tex]\( b \)[/tex] by dividing consecutive function values: [tex]\( b = \frac{f(x+1)}{f(x)} \)[/tex].

[tex]\[ f(1) = \][/tex]
[tex]\[ \square \][/tex]
[tex]\[ \square \text{ and } f(2) = \square \][/tex]
[tex]\[ \text{so} \ b = \frac{f(2)}{f(1)} = \square \][/tex]

Step 2 - Use the exponential function equation [tex]\( f(x) = ab^x \)[/tex] to solve for [tex]\( a \)[/tex].

Choosing [tex]\(\square\)[/tex],
[tex]\[ x = \square, \quad f(x) = \square \][/tex]
[tex]\[ \text{and } b = \square. \][/tex]

Substituting, you obtain:
[tex]\[ f(x) = ab^x \][/tex]

Choosing [tex]\(\square\)[/tex],
[tex]\[ x = \square, \quad f(x) = \square, \][/tex]
[tex]\[ \text{and } b = \square. \][/tex]

Substituting, you obtain:
[tex]\[ f(x) = ab^x \][/tex]

Step 3 - Write the exponential function [tex]\( f(x) = ab^x \)[/tex] (substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex]).

[tex]\[ a = \][/tex]
[tex]\[ \square \][/tex]
[tex]\[ b = \][/tex]
[tex]\[ \square \][/tex]
[tex]\[ \text{so} \ f(x) = ab^x \][/tex]

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This formatted version maintains clarity and step-by-step guidance to help students understand the process of writing an exponential function from given points.



Answer :

GinnyAnswer
To write the exponential function that passes through the points (1,6) and (2,12), we will follow a detailed step-by-step solution:

### Step 1: Solve for [tex]\( b \)[/tex] by dividing consecutive function values
The function values at the given points are:

[tex]\[ f(1) = 6 \][/tex]
[tex]\[ f(2) = 12 \][/tex]

We need to solve for [tex]\( b \)[/tex] using the relationship:
[tex]\[ b = \frac{f(x+1)}{f(x)} \][/tex]
Substituting the given values:
[tex]\[ b = \frac{f(2)}{f(1)} = \frac{12}{6} = 2 \][/tex]

So, [tex]\( b = 2 \)[/tex].

### Step 2: Use the exponential function equation [tex]\( f(x) = a b^x \)[/tex] to solve for [tex]\( a \)[/tex]

Using the point [tex]\((1, 6)\)[/tex]:
[tex]\[ f(x) = a b^x \][/tex]
[tex]\[ f(1) = a b^1 \][/tex]
[tex]\[ 6 = a \cdot 2^1 \][/tex]
[tex]\[ 6 = 2a \][/tex]
[tex]\[ a = \frac{6}{2} = 3 \][/tex]

Thus, [tex]\( a = 3 \)[/tex].

### Step 3: Write the exponential function [tex]\( f(x) = a b^x \)[/tex]
Substitute [tex]\( a = 3 \)[/tex] and [tex]\( b = 2 \)[/tex] into the exponential function equation:

So, the exponential function is:
[tex]\[ f(x) = 3 \cdot 2^x \][/tex]

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