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.
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.
To compare the parabola defined by 𝑦 = −4(𝑥 + 3)² − 7 to the standard parabola 𝑦 = 𝑥², we need to identify the transformations applied to the standard parabola.
The standard parabola 𝑦 = 𝑥² has the following features:
1. Vertex: (0, 0)
2. Axis of Symmetry: x = 0 (the y-axis)
3. Opens: upwards
Now let's analyze the given equation: 𝑦 = −4(𝑥 + 3)² − 7.
Transformation 1: Reflection over the x-axis
The negative sign outside the brackets −4(𝑥 + 3) indicates a reflection of the standard parabola across the x-axis. This means the new parabola will open downwards instead of upwards.
Transformation 2: Horizontal Translation
The expression 𝑥 + 3 inside the brackets 𝑦 = −4(𝑥 + 3)² indicates a horizontal translation of 3 units to the left. This means the vertex of the new parabola will shift horizontally 3 units to the left compared to the standard parabola.
Transformation 3: Vertical Translation
The constant term - 7 indicates a vertical translation of 7 units downwards. This means the new parabola's vertex will shift 7 units downward compared to the standard parabola's vertex.
To summarize: the parabola defined by 𝑦 = −4(𝑥 + 3)² − 7 is a downward reflection (compared to the standard parabola) with a horizontal translation of 3 units to the left and a vertical translation of 7 units downward.