To rewrite the transformations of the parent function \( y = x^2 \) in the form \( y = a(x - h)^2 + k \), we will apply the necessary transformations for each case.
A. The graph has been shifted to the left 1 and up 5
- Shift left 1: This means we replace \( x \) with \( (x + 1) \).
- Shift up 5: This means we add 5 to the entire function.
Putting it together: \[ y = (x + 1)^2 + 5 \]
B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3
- Reflection over the x-axis: This changes the sign of the function, giving us \( -x^2 \).
- Stretched vertically by a factor of 3: We multiply by 3, getting \( -3x^2 \).
To write this in the desired form, we factor it: \[ y = -3(x - 0)^2 + 0 \]
So: \[ y = -3(x - 0)^2 + 0 \]
C. The graph has been compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6
- Compressed vertically by a factor of \( \frac{1}{2} \): This means we multiply the function by \( \frac{1}{2} \), resulting in \( \frac{1}{2}x^2 \).
- Shifted right 6: We replace \( x \) with \( (x - 6) \).
Putting it together: \[ y = \frac{1}{2}(x - 6)^2 \]
Summary of Transformations
A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)
These are your transformed equations in the requested form.
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