Think about the standard parabola defined by 𝑦 = 𝑥2. How does the parabola

Question

defined by 𝑦 = −4(𝑥 + 3)2 − 7 compare to the standard parabola? Describe all of
the transformations. Then, tell the steps draw a reasonable sketch of both parabolas

Answer 1

The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a vertical compression and reflection of the standard parabola.

Transformations:
1. Reflection across the x-axis (negative coefficient of 𝑦)
2. Horizontal translation 3 units to the left (opposite of the sign in the parentheses)
3. Vertical compression by a factor of 4 (coefficient in front of the squared term)
4. Vertical translation 7 units downward

Steps to draw a reasonable sketch of both parabolas:
1. Plot the vertex of 𝑦 = 𝑥2 at (0, 0)
2. Plot the vertex of 𝑦 = −4(𝑥 + 3)2 − 7, which is shifted 3 units to the left and 7 units downward from the vertex of 𝑦 = 𝑥2, at (-3, -7)
3. Sketch the vertical compression of the parabola by making the arms of the parabola more narrow than the standard parabola
4. Sketch the reflection across the x-axis by making the parabola open downward instead of upward
5. Connect the curve smoothly and label both parabolas.

Here is an example sketch:


Answer 2

BZZZZT! But thanks for playing.

3. Vertical dilation by a factor of 4

Answer 3

My apologies, you are correct. The correct transformation should be vertical dilation by a factor of 4, not compression. Thank you for bringing it to my attention!