QuestionFind a power function passing through the points $(1,3)$ and $(4,13)$ .
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Studdy Solution

STEP 1

1. A power function is of the form $y = ax^b$ .
2. The function passes through the points $(1, 3)$ and $(4, 13)$ .

STEP 2

1. Set up equations using the given points.
2. Solve the system of equations to find the constants $a$ and $b$ .
3. Write the power function.

STEP 3

Substitute the point $(1, 3)$ into the power function equation $y = ax^b$ .
This gives us the equation: $3 = a(1)^b$

STEP 4

Simplify the equation from STEP_1. Since $1^b = 1$ , the equation becomes: $3 = a$

STEP 5

Substitute the point $(4, 13)$ into the power function equation $y = ax^b$ .
This gives us the equation: $13 = a(4)^b$

STEP 6

Use the value of $a$ found in STEP_2, which is $a = 3$ , and substitute it into the equation from STEP_3: $13 = 3(4)^b$

STEP 7

Solve for $b$ in the equation $13 = 3 \times 4^b$ .
Divide both sides by 3: $\frac{13}{3} = 4^b$
Take the logarithm of both sides to solve for $b$ : $\log\left(\frac{13}{3}\right) = b \log(4)$
Solve for $b$ : $b = \frac{\log\left(\frac{13}{3}\right)}{\log(4)}$

STEP 8

Now that we have $a = 3$ and $b = \frac{\log\left(\frac{13}{3}\right)}{\log(4)}$ , the power function is: $y = 3x^{\frac{\log\left(\frac{13}{3}\right)}{\log(4)}}$
The power function passing through the points $(1, 3)$ and $(4, 13)$ is:
$y = 3x^{\frac{\log\left(\frac{13}{3}\right)}{\log(4)}}$

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