Certainly! To write an equation for the given exponential function, we need to follow some steps systematically.
The general form of an exponential function is:
[tex]\[ y = a \cdot b^x \][/tex]
Here, [tex]\(a\)[/tex] represents the initial value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex], and [tex]\(b\)[/tex] is the base of the exponential function which represents the constant ratio between successive [tex]\(y\)[/tex]-values.
Let's break down the process step by step:
1. Identify the initial value [tex]\( a \)[/tex]:
The initial value [tex]\( a \)[/tex] can be directly taken from the table where [tex]\(x = 0\)[/tex].
From the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 6 \\
\hline
\end{array}
\][/tex]
So, [tex]\( a = 6 \)[/tex].
2. Determine the base [tex]\( b \)[/tex]:
To find the base [tex]\( b \)[/tex], we'll use the ratio of [tex]\( y \)[/tex]-values for successive values of [tex]\( x \)[/tex].
For [tex]\(x = 1\)[/tex] and [tex]\(x = 2\)[/tex]:
[tex]\[
\frac{y(2)}{y(1)} = \frac{24}{12} = 2
\][/tex]
The base [tex]\( b \)[/tex] is 2.
3. Form the equation:
Now that we know [tex]\( a = 6 \)[/tex] and [tex]\( b = 2 \)[/tex], we can write the equation of the exponential function as:
[tex]\[
y = 6 \cdot 2^x
\][/tex]
Let's confirm the equation with all given points from the table:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 6 \cdot 2^{-1} = 6 \cdot \frac{1}{2} = 3
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 6 \cdot 2^0 = 6 \cdot 1 = 6
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 6 \cdot 2^1 = 6 \cdot 2 = 12
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6 \cdot 2^2 = 6 \cdot 4 = 24
\][/tex]
Each calculated [tex]\( y \)[/tex]-value matches the corresponding [tex]\( y \)[/tex]-value in the table, verifying our equation.
Thus, the equation for the given exponential function is:
[tex]\[
y = 6 \cdot 2^x
\][/tex]
