Think about the standard parabola defined by 𝑦 = 𝑥2. How does the parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 compare to the standard parabola? Describe all of the transformations.

Answer 1

The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a transformation of the standard parabola 𝑦 = 𝑥2.

First, the negative coefficient (-4) reflects the parabola across the 𝑥-axis, which means it is now upside down compared to the standard parabola.

Next, the factor of 4 stretching factor is combined with the negative sign to vertically compress the parabola, making it narrower than the standard parabola.

Then, the entire parabola is shifted 3 units to the left with the addition of the term (𝑥 + 3), moving it away from the origin.

Finally, the additional constant term -7 shifts the parabola downwards by 7 units.

Overall, the parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a vertically compressed, inverted version of the standard parabola, shifted 3 units to the left and downwards 7 units.