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Mod 8, 10, 811
Question 2 of 10 (1 point) | Question Attempt: 1 of 1
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The table of ordered pairs 
(x,y) gives an exponential function. Write an equation for the function.





x

y


-1
24


0
6


1

(3)/(2)


2

(3)/(8)

Mod 88, 1010, 811811\newlineQuestion 22 of 1010 (11 point) | Question Attempt: 11 of 11\newline11\newline22\newline33\newline44\newline55\newlineThe table of ordered pairs (x,y) (x, y) gives an exponential function. Write an equation for the function.\newline\begin{tabular}{|c|c|}\newline\hlinex x & y y \\\newline\hline1-1 & 2424 \\\newline\hline 00 & 66 \\\newline\hline 11 & 32 \frac{3}{2} \\\newline\hline 22 & 38 \frac{3}{8} \\\newline\hline\newline\end{tabular}

Full solution

Q. Mod 88, 1010, 811811\newlineQuestion 22 of 1010 (11 point) | Question Attempt: 11 of 11\newline11\newline22\newline33\newline44\newline55\newlineThe table of ordered pairs (x,y) (x, y) gives an exponential function. Write an equation for the function.\newline\begin{tabular}{|c|c|}\newline\hlinex x & y y \\\newline\hline1-1 & 2424 \\\newline\hline 00 & 66 \\\newline\hline 11 & 32 \frac{3}{2} \\\newline\hline 22 & 38 \frac{3}{8} \\\newline\hline\newline\end{tabular}
  1. Observe Decreasing YY-values: Notice that the yy-values are decreasing as the xx-values increase, which is typical for an exponential decay function.
  2. Find Base of Function: To find the base of the exponential function, look at the y-values when xx increases by 11. Going from x=1x = -1 to x=0x = 0, yy decreases from 2424 to 66.
  3. Calculate Base Value: Calculate the base bb by dividing the y-value at x=0x = 0 by the y-value at x=1x = -1: b=624=14b = \frac{6}{24} = \frac{1}{4}.
  4. Confirm Base Calculation: Check the base with another pair of points. Going from x=0x = 0 to x=1x = 1, yy decreases from 66 to 32\frac{3}{2}. Calculate b=326=14b = \frac{\frac{3}{2}}{6} = \frac{1}{4} again, which confirms our base.
  5. Write Exponential Function: Now, write the general form of the exponential function: y=abxy = a \cdot b^x. We already know b=14b = \frac{1}{4}. To find aa, use the point (0,6)(0, 6) because any number to the power of 00 is 11.
  6. Determine Value of aa: Plug x=0x = 0 and y=6y = 6 into the equation: 6=a×(14)06 = a \times (\frac{1}{4})^0. Since (14)0=1(\frac{1}{4})^0 = 1, it follows that a=6a = 6.
  7. Final Exponential Equation: Write the final equation of the exponential function: y=6×(14)xy = 6 \times \left(\frac{1}{4}\right)^x.

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