Discover new knowledge and insights with IDNLearn.com's extensive Q&A database. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
Sagot :
To write an exponential function [tex]\( y = a \cdot b^x \)[/tex] that represents the given data in the table, let's follow a clear, step-by-step approach:
1. Identify Constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- First, observe the values of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0:
[tex]\[ \begin{align*} x & = 0 \\ y & = 1 \\ \end{align*} \][/tex]
Substituting [tex]\( x = 0 \)[/tex] into the exponential function [tex]\( y = a \cdot b^0 \)[/tex], we get:
[tex]\[ y = a \cdot 1 = a \][/tex]
Therefore, [tex]\( a = 1 \)[/tex].
2. Determine the Base [tex]\( b \)[/tex]:
- Use another pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values to find [tex]\( b \)[/tex]. Consider [tex]\( x = 1 \)[/tex]:
[tex]\[ \begin{align*} x & = 1 \\ y & = 3 \\ \end{align*} \][/tex]
Substitute these values into the function:
[tex]\[ 3 = 1 \cdot b^1 \implies b = 3 \][/tex]
Therefore, [tex]\( b = 3 \)[/tex].
3. Formulate the Exponential Function:
Using the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] found:
[tex]\[ y = 1 \cdot 3^x \quad \text{or simply} \quad y = 3^x \][/tex]
4. Verification:
To ensure our function [tex]\( y = 3^x \)[/tex] fits all given points in the table, let’s check:
[tex]\[ \begin{array}{cc} x & y = 3^x \\ \hline -2 & 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \\ -1 & 3^{-1} = \frac{1}{3} \\ 0 & 3^0 = 1 \\ 1 & 3^1 = 3 \\ 2 & 3^2 = 9 \\ \end{array} \][/tex]
The function [tex]\( y = 3^x \)[/tex] consistently matches all given [tex]\( y \)[/tex] values from the table.
Therefore, the exponential function that represents the data in the table is:
[tex]\[ y = 3^x \][/tex]
1. Identify Constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- First, observe the values of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0:
[tex]\[ \begin{align*} x & = 0 \\ y & = 1 \\ \end{align*} \][/tex]
Substituting [tex]\( x = 0 \)[/tex] into the exponential function [tex]\( y = a \cdot b^0 \)[/tex], we get:
[tex]\[ y = a \cdot 1 = a \][/tex]
Therefore, [tex]\( a = 1 \)[/tex].
2. Determine the Base [tex]\( b \)[/tex]:
- Use another pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values to find [tex]\( b \)[/tex]. Consider [tex]\( x = 1 \)[/tex]:
[tex]\[ \begin{align*} x & = 1 \\ y & = 3 \\ \end{align*} \][/tex]
Substitute these values into the function:
[tex]\[ 3 = 1 \cdot b^1 \implies b = 3 \][/tex]
Therefore, [tex]\( b = 3 \)[/tex].
3. Formulate the Exponential Function:
Using the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] found:
[tex]\[ y = 1 \cdot 3^x \quad \text{or simply} \quad y = 3^x \][/tex]
4. Verification:
To ensure our function [tex]\( y = 3^x \)[/tex] fits all given points in the table, let’s check:
[tex]\[ \begin{array}{cc} x & y = 3^x \\ \hline -2 & 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \\ -1 & 3^{-1} = \frac{1}{3} \\ 0 & 3^0 = 1 \\ 1 & 3^1 = 3 \\ 2 & 3^2 = 9 \\ \end{array} \][/tex]
The function [tex]\( y = 3^x \)[/tex] consistently matches all given [tex]\( y \)[/tex] values from the table.
Therefore, the exponential function that represents the data in the table is:
[tex]\[ y = 3^x \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.