QuestionDetermine the linear function $f(x)$ with $f(-3)=2$ and $f(4)=16$ .

Studdy Solution

STEP 1

Assumptions1. The function is linear, which means it can be represented in the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. . We are given two points on the line $(-3,)$ and $(4,16)$ .

STEP 2

We can find the slope of the line using the formula for slope, which is the change in $y$ divided by the change in $x$ .
$m = \frac{y2 - y1}{x2 - x1}$

STEP 3

Substitute the given points into the slope formula.
$m = \frac{16 -2}{ - (-3)}$

STEP 4

Calculate the slope.
$m = \frac{14}{7} =2$

STEP 5

Now that we have the slope, we can find the y-intercept $b$ by rearranging the equation of the line and plugging in the values for one of the points and the slope.
$b = y - mx$

STEP 6

Substitute the values for $m$ , $x$ , and $y$ from one of the points into the equation for $b$ . Let's use the point $(-3,2)$ .
$b =2 -2(-3)$

STEP 7

Calculate the y-intercept.
$b =2 +6 =$

STEP 8

Now that we have both the slope and the y-intercept, we can write the equation of the line.
$f(x) = mx + b$

STEP 9

Substitute the values for $m$ and $b$ into the equation.
$f(x) =2x +8$ So, the formula for the linear function is $f(x) =2x +8$ .

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QuestionDetermine the linear function $f(x)$ with $f(-3)=2$ and $f(4)=16$ .

Studdy Solution

STEP 1

Assumptions1. The function is linear, which means it can be represented in the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. . We are given two points on the line $(-3,)$ and $(4,16)$ .

STEP 2

We can find the slope of the line using the formula for slope, which is the change in $y$ divided by the change in $x$ .
$m = \frac{y2 - y1}{x2 - x1}$

STEP 3

Substitute the given points into the slope formula.
$m = \frac{16 -2}{ - (-3)}$

STEP 4

Calculate the slope.
$m = \frac{14}{7} =2$

STEP 5

Now that we have the slope, we can find the y-intercept $b$ by rearranging the equation of the line and plugging in the values for one of the points and the slope.
$b = y - mx$

STEP 6

Substitute the values for $m$ , $x$ , and $y$ from one of the points into the equation for $b$ . Let's use the point $(-3,2)$ .
$b =2 -2(-3)$

STEP 7

Calculate the y-intercept.
$b =2 +6 =$

STEP 8

Now that we have both the slope and the y-intercept, we can write the equation of the line.
$f(x) = mx + b$

STEP 9

Substitute the values for $m$ and $b$ into the equation.
$f(x) =2x +8$ So, the formula for the linear function is $f(x) =2x +8$ .

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QuestionDetermine the linear function f(x)f(x)f(x) with f(−3)=2f(-3)=2f(−3)=2 and f(4)=16f(4)=16f(4)=16.

Studdy Solution

STEP 1

Assumptions1. The function is linear, which means it can be represented in the form f(x)=mx+bf(x) = mx + bf(x)=mx+b, where mmm is the slope and bbb is the y-intercept. . We are given two points on the line (−3,)(-3,)(−3,) and (4,16)(4,16)(4,16).

STEP 2

We can find the slope of the line using the formula for slope, which is the change in yyy divided by the change in xxx.m=y2−y1x2−x1m = \frac{y2 - y1}{x2 - x1}m=x2−x1y2−y1​

STEP 3

Substitute the given points into the slope formula.m=16−2−(−3)m = \frac{16 -2}{ - (-3)}m=−(−3)16−2​

STEP 4

Calculate the slope.m=147=2m = \frac{14}{7} =2m=714​=2

STEP 5

Now that we have the slope, we can find the y-intercept bbb by rearranging the equation of the line and plugging in the values for one of the points and the slope.b=y−mxb = y - mxb=y−mx

STEP 6

Substitute the values for mmm, xxx, and yyy from one of the points into the equation for bbb. Let's use the point (−3,2)(-3,2)(−3,2).b=2−2(−3)b =2 -2(-3)b=2−2(−3)

STEP 7

Calculate the y-intercept.b=2+6=b =2 +6 =b=2+6=

STEP 8

Now that we have both the slope and the y-intercept, we can write the equation of the line.f(x)=mx+bf(x) = mx + bf(x)=mx+b

STEP 9

Substitute the values for mmm and bbb into the equation.f(x)=2x+8f(x) =2x +8f(x)=2x+8So, the formula for the linear function is f(x)=2x+8f(x) =2x +8f(x)=2x+8.

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QuestionDetermine the linear function $f(x)$ with $f(-3)=2$ and $f(4)=16$ .

Assumptions1. The function is linear, which means it can be represented in the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. . We are given two points on the line $(-3,)$ and $(4,16)$ .

We can find the slope of the line using the formula for slope, which is the change in $y$ divided by the change in $x$ .
$m = \frac{y2 - y1}{x2 - x1}$

Substitute the given points into the slope formula.
$m = \frac{16 -2}{ - (-3)}$

Calculate the slope.
$m = \frac{14}{7} =2$

Now that we have the slope, we can find the y-intercept $b$ by rearranging the equation of the line and plugging in the values for one of the points and the slope.
$b = y - mx$

Substitute the values for $m$ , $x$ , and $y$ from one of the points into the equation for $b$ . Let's use the point $(-3,2)$ .
$b =2 -2(-3)$

Calculate the y-intercept.
$b =2 +6 =$

Now that we have both the slope and the y-intercept, we can write the equation of the line.
$f(x) = mx + b$

Substitute the values for $m$ and $b$ into the equation.
$f(x) =2x +8$ So, the formula for the linear function is $f(x) =2x +8$ .