Solved on Sep 21, 2023

Find power function $y=a \times b^x$ and linear function $y=mx+c$ to model data set $(x, y)$ . Power function coefficients $a=4.287$ , $b=1.204$ . Determine which function better fits the data.

STEP 1

Assumptions1. The given data set is\begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline1 &5 \\ \hline &9 \\ \hline3 &13 \\ \hline4 &21 \\ \hline5 &31 \\ \hline6 &45 \\ \hline\end{tabular}
. The power function is in the form $y=a \times b^x$ , where $a$ and $b$ are constants.
3. The linear function is in the form $y=mx+c$ , where $m$ and $c$ are constants.
4. The user has already found a power function $y=4.287 \times1.204^x$ .

STEP 2

To find the linear function that models the data, we first need to calculate the slope $m$ . The slope is calculated by taking the difference in $y$ values divided by the difference in $x$ values between two points.
$m = \frac{y2 - y1}{x2 - x1}$

STEP 3

Choose two points from the data set. For simplicity, we'll use the first two points $(1,5)$ and $(2,9)$ .

STEP 4

Substitute the values of the points into the slope formula.
$m = \frac{9 -}{2 -1}$

STEP 5

Calculate the slope $m$ .
$m = \frac{9 -5}{2 -1} =4$

STEP 6

Now that we have the slope, we can find the y-intercept $c$ . The y-intercept is the $y$ value when $x=0$ . We can find this by rearranging the linear function formula to solve for $c$ .
$c = y - mx$

STEP 7

Substitute the slope $m$ and one point $(x,y)$ into the y-intercept formula. We'll use the point $(1,5)$ .
$c =5 -4 \times1$

STEP 8

Calculate the y-intercept $c$ .
$c =5 -4 \times1 =1$

STEP 9

Now that we have the slope $m$ and the y-intercept $c$ , we can write the linear function that models the data.
$y = mx + c$

STEP 10

Substitute the values of $m$ and $c$ into the linear function.
$y =4x +$ The linear function that models the data is $y =4x +$ .

STEP 11

While the user has found a power function $y=4.287 \times.204^x$ , it's important to note that finding a power function typically requires logarithmic regression to determine the coefficients. Therefore, the power function might not be accurate.

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Find power function $y=a \times b^x$ and linear function $y=mx+c$ to model data set $(x, y)$ . Power function coefficients $a=4.287$ , $b=1.204$ . Determine which function better fits the data.

STEP 1

STEP 2

To find the linear function that models the data, we first need to calculate the slope $m$ . The slope is calculated by taking the difference in $y$ values divided by the difference in $x$ values between two points.
$m = \frac{y2 - y1}{x2 - x1}$

STEP 3

Choose two points from the data set. For simplicity, we'll use the first two points $(1,5)$ and $(2,9)$ .

STEP 4

Substitute the values of the points into the slope formula.
$m = \frac{9 -}{2 -1}$

STEP 5

Calculate the slope $m$ .
$m = \frac{9 -5}{2 -1} =4$

STEP 6

Now that we have the slope, we can find the y-intercept $c$ . The y-intercept is the $y$ value when $x=0$ . We can find this by rearranging the linear function formula to solve for $c$ .
$c = y - mx$

STEP 7

Substitute the slope $m$ and one point $(x,y)$ into the y-intercept formula. We'll use the point $(1,5)$ .
$c =5 -4 \times1$

STEP 8

Calculate the y-intercept $c$ .
$c =5 -4 \times1 =1$

STEP 9

Now that we have the slope $m$ and the y-intercept $c$ , we can write the linear function that models the data.
$y = mx + c$

STEP 10

Substitute the values of $m$ and $c$ into the linear function.
$y =4x +$ The linear function that models the data is $y =4x +$ .

STEP 11

While the user has found a power function $y=4.287 \times.204^x$ , it's important to note that finding a power function typically requires logarithmic regression to determine the coefficients. Therefore, the power function might not be accurate.

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Find power function y=a×bxy=a \times b^xy=a×bx and linear function y=mx+cy=mx+cy=mx+c to model data set (x,y)(x, y)(x,y). Power function coefficients a=4.287a=4.287a=4.287, b=1.204b=1.204b=1.204. Determine which function better fits the data.

STEP 1

STEP 2

To find the linear function that models the data, we first need to calculate the slope mmm. The slope is calculated by taking the difference in yyy values divided by the difference in xxx values between two points.m=y2−y1x2−x1m = \frac{y2 - y1}{x2 - x1}m=x2−x1y2−y1​

STEP 3

Choose two points from the data set. For simplicity, we'll use the first two points (1,5)(1,5)(1,5) and (2,9)(2,9)(2,9).

STEP 4

Substitute the values of the points into the slope formula.m=9−2−1m = \frac{9 -}{2 -1}m=2−19−​

STEP 5

Calculate the slope mmm.m=9−52−1=4m = \frac{9 -5}{2 -1} =4m=2−19−5​=4

STEP 6

Now that we have the slope, we can find the y-intercept ccc. The y-intercept is the yyy value when x=0x=0x=0. We can find this by rearranging the linear function formula to solve for ccc.c=y−mxc = y - mxc=y−mx

STEP 7

Substitute the slope mmm and one point (x,y)(x,y)(x,y) into the y-intercept formula. We'll use the point (1,5)(1,5)(1,5).c=5−4×1c =5 -4 \times1c=5−4×1

STEP 8

Calculate the y-intercept ccc.c=5−4×1=1c =5 -4 \times1 =1c=5−4×1=1

STEP 9

Now that we have the slope mmm and the y-intercept ccc, we can write the linear function that models the data.y=mx+cy = mx + cy=mx+c

STEP 10

Substitute the values of mmm and ccc into the linear function.y=4x+y =4x +y=4x+The linear function that models the data is y=4x+y =4x +y=4x+.

STEP 11

While the user has found a power function y=4.287×.204xy=4.287 \times.204^xy=4.287×.204x, it's important to note that finding a power function typically requires logarithmic regression to determine the coefficients. Therefore, the power function might not be accurate.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find power function $y=a \times b^x$ and linear function $y=mx+c$ to model data set $(x, y)$ . Power function coefficients $a=4.287$ , $b=1.204$ . Determine which function better fits the data.

To find the linear function that models the data, we first need to calculate the slope $m$ . The slope is calculated by taking the difference in $y$ values divided by the difference in $x$ values between two points.
$m = \frac{y2 - y1}{x2 - x1}$

Choose two points from the data set. For simplicity, we'll use the first two points $(1,5)$ and $(2,9)$ .

Substitute the values of the points into the slope formula.
$m = \frac{9 -}{2 -1}$

Calculate the slope $m$ .
$m = \frac{9 -5}{2 -1} =4$

Now that we have the slope, we can find the y-intercept $c$ . The y-intercept is the $y$ value when $x=0$ . We can find this by rearranging the linear function formula to solve for $c$ .
$c = y - mx$

Substitute the slope $m$ and one point $(x,y)$ into the y-intercept formula. We'll use the point $(1,5)$ .
$c =5 -4 \times1$

Calculate the y-intercept $c$ .
$c =5 -4 \times1 =1$

Now that we have the slope $m$ and the y-intercept $c$ , we can write the linear function that models the data.
$y = mx + c$

Substitute the values of $m$ and $c$ into the linear function.
$y =4x +$ The linear function that models the data is $y =4x +$ .

While the user has found a power function $y=4.287 \times.204^x$ , it's important to note that finding a power function typically requires logarithmic regression to determine the coefficients. Therefore, the power function might not be accurate.