Question
I'm studying for my math exam and I need to find the equation of a parabola with vertex (-8, 10) that passes through (4, 16). Can you find a similar problem on the Q&A forum that provides a step-by-step solution to finding the equation of a parabola given its vertex and a point, and summarize the solution process for me?
Answers
GPT-4o mini
Certainly! While I can't browse the Q&A forum for you, I can definitely guide you through a similar problem step-by-step in order to help you with your math exam preparation.
**Problem: Find the equation of a parabola with vertex (-8, 10) that passes through the point (4, 16).**
### Step 1: Use the vertex form of a parabola
The vertex form of a parabola is given by the equation:
\[
y = a(x - h)^2 + k
\]
where (h, k) is the vertex of the parabola. In this case, our vertex is (-8, 10), which means \( h = -8 \) and \( k = 10 \).
### Step 2: Substitute the vertex into the vertex form
Substituting the vertex into our equation, we get:
\[
y = a(x + 8)^2 + 10
\]
### Step 3: Use the given point to find 'a'
Now, we need to use the point (4, 16) to find the value of \( a \). Substitute \( x = 4 \) and \( y = 16 \) into the equation:
\[
16 = a(4 + 8)^2 + 10
\]
This simplifies to:
\[
16 = a(12)^2 + 10
\]
\[
16 = 144a + 10
\]
### Step 4: Solve for 'a'
Next, isolate \( a \) by first subtracting 10 from both sides:
\[
6 = 144a
\]
Now divide both sides by 144:
\[
a = \frac{6}{144} = \frac{1}{24}
\]
### Step 5: Write the final equation
Now that we have found \( a \), we can substitute it back into the vertex form of the parabola:
\[
y = \frac{1}{24}(x + 8)^2 + 10
\]
### Summary
The final equation of the parabola is:
\[
y = \frac{1}{24}(x + 8)^2 + 10
\]
By following these steps, you can find the equation of a parabola given its vertex and a point. Be sure to always start with the vertex form and use the provided point to solve for the coefficient \( a \). Good luck with your exam!
**Problem: Find the equation of a parabola with vertex (-8, 10) that passes through the point (4, 16).**
### Step 1: Use the vertex form of a parabola
The vertex form of a parabola is given by the equation:
\[
y = a(x - h)^2 + k
\]
where (h, k) is the vertex of the parabola. In this case, our vertex is (-8, 10), which means \( h = -8 \) and \( k = 10 \).
### Step 2: Substitute the vertex into the vertex form
Substituting the vertex into our equation, we get:
\[
y = a(x + 8)^2 + 10
\]
### Step 3: Use the given point to find 'a'
Now, we need to use the point (4, 16) to find the value of \( a \). Substitute \( x = 4 \) and \( y = 16 \) into the equation:
\[
16 = a(4 + 8)^2 + 10
\]
This simplifies to:
\[
16 = a(12)^2 + 10
\]
\[
16 = 144a + 10
\]
### Step 4: Solve for 'a'
Next, isolate \( a \) by first subtracting 10 from both sides:
\[
6 = 144a
\]
Now divide both sides by 144:
\[
a = \frac{6}{144} = \frac{1}{24}
\]
### Step 5: Write the final equation
Now that we have found \( a \), we can substitute it back into the vertex form of the parabola:
\[
y = \frac{1}{24}(x + 8)^2 + 10
\]
### Summary
The final equation of the parabola is:
\[
y = \frac{1}{24}(x + 8)^2 + 10
\]
By following these steps, you can find the equation of a parabola given its vertex and a point. Be sure to always start with the vertex form and use the provided point to solve for the coefficient \( a \). Good luck with your exam!