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Consider a function with the values shown in the table below:

[tex]\[
\begin{array}{ccccc}
x & 1 & 6 & 8 & 12 \\
y & 12 & 27 & 33 & 45 \\
\end{array}
\][/tex]

Find the average rate of change between [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex].

Average rate of change [tex]\( = \)[/tex] [tex]\(\square\)[/tex]



Answer :

GinnyAnswer
Sure! To find the average rate of change between [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex], we follow these steps:

1. Identify the values of [tex]\( y \)[/tex] corresponding to [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 12 \)[/tex]
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 27 \)[/tex]

2. Calculate the change in [tex]\( y \)[/tex] ([tex]\( \Delta y \)[/tex]):
[tex]\[ \Delta y = y(6) - y(1) = 27 - 12 = 15 \][/tex]

3. Calculate the change in [tex]\( x \)[/tex] ([tex]\( \Delta x \)[/tex]):
[tex]\[ \Delta x = 6 - 1 = 5 \][/tex]

4. Find the average rate of change:
[tex]\[ \text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{15}{5} = 3.0 \][/tex]

Therefore, the average rate of change between [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex] is [tex]\( 3.0 \)[/tex].

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