Math

Question Describe the transformations that transform y=x2y=x^2 into y=(x7)2+4y=-(x-7)^2+4.

Studdy Solution

STEP 1

Assumptions
1. The original parabola is y=x2y = x^{2}.
2. The transformed parabola is y=(x7)2+4y = -(x - 7)^{2} + 4.
3. We are looking for the transformations that turn the original parabola into the transformed one.

STEP 2

Identify the horizontal shift.
The term (x7)(x - 7) in the transformed equation indicates a horizontal shift. The direction of the shift is determined by the sign inside the parentheses. A positive sign would indicate a shift to the left, while a negative sign indicates a shift to the right.

STEP 3

State the horizontal shift.
Since we have (x7)(x - 7), this means the parabola has been shifted 7 units to the right.

STEP 4

Identify the vertical shift.
The term +4+4 at the end of the transformed equation indicates a vertical shift. The direction of the shift is determined by the sign of the number. A positive number indicates an upward shift, while a negative number indicates a downward shift.

STEP 5

State the vertical shift.
Since we have +4+4, this means the parabola has been shifted 4 units upward.

STEP 6

Identify the reflection.
The negative sign in front of the squared term (x7)2-(x - 7)^{2} indicates a reflection across the x-axis.

STEP 7

State the reflection.
The negative sign means the parabola has been reflected across the x-axis.

STEP 8

Combine all the transformations.
The parabola y=x2y = x^{2} has undergone the following transformations to become y=(x7)2+4y = -(x - 7)^{2} + 4:
1. A horizontal shift 7 units to the right.
2. A vertical shift 4 units upward.
3. A reflection across the x-axis.

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