Question
What is the equation of a parabola that was shifted right 8 units and up 1 unit, and had a stretch factor of 4
Answers
GPT-4o mini
To find the equation of a parabola that has been transformed, we can start from the standard form of a parabola, which is \(y = ax^2\), where \(a\) is the stretch factor.
Given the transformations:
1. **Shift right by 8 units:** This means we replace \(x\) with \((x - 8)\).
2. **Shift up by 1 unit:** This means we add 1 to the entire equation.
3. **Stretch factor of 4:** This means \(a = 4\).
Starting from the base equation \(y = ax^2\) with the stretch factor of 4, we have:
\[
y = 4x^2
\]
Now, applying the horizontal shift first (right 8 units):
\[
y = 4(x - 8)^2
\]
Next, applying the vertical shift (up 1 unit):
\[
y = 4(x - 8)^2 + 1
\]
So, the final equation of the parabola after all transformations is:
\[
y = 4(x - 8)^2 + 1
\]
Given the transformations:
1. **Shift right by 8 units:** This means we replace \(x\) with \((x - 8)\).
2. **Shift up by 1 unit:** This means we add 1 to the entire equation.
3. **Stretch factor of 4:** This means \(a = 4\).
Starting from the base equation \(y = ax^2\) with the stretch factor of 4, we have:
\[
y = 4x^2
\]
Now, applying the horizontal shift first (right 8 units):
\[
y = 4(x - 8)^2
\]
Next, applying the vertical shift (up 1 unit):
\[
y = 4(x - 8)^2 + 1
\]
So, the final equation of the parabola after all transformations is:
\[
y = 4(x - 8)^2 + 1
\]