9. 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. Then, draw a reasonable sketch of both parabolas.
The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is the standard parabola 𝑦 = 𝑥2 that has undergone a few transformations.
First, the entire graph has been shifted three units to the left (because of the (𝑥 + 3) term inside the parentheses).
Next, the parabola has been stretched vertically by a factor of 4 (because of the coefficient -4 in front of the parentheses).
Finally, the parabola has been shifted seven units downward (because of the -7 added at the end).
A rough sketch of the two parabolas would look like this:
To compare the parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 to the standard parabola 𝑦 = 𝑥2, we need to analyze the transformations:
1. Vertical translation: The standard parabola 𝑦 = 𝑥2 does not have any vertical translation. However, the parabola 𝑦 = −4(𝑥 + 3)2 − 7 is shifted downward by 7 units.
2. Horizontal translation: The standard parabola 𝑦 = 𝑥2 does not have any horizontal translation. But in the equation 𝑦 = −4(𝑥 + 3)2 − 7, there is a horizontal translation of 3 units to the left.
3. Vertical stretch/compression: The coefficient -4 in 𝑦 = −4(𝑥 + 3)2 − 7 represents a vertical compression. The parabola is vertically compressed by a factor of 4 compared to the standard parabola.
Now, let's draw a sketch of both parabolas:
First, let's draw the standard parabola 𝑦 = 𝑥2:
- It is symmetric about the y-axis, passing through the origin (0,0).
- It opens upward.
- It gradually becomes steeper as x increases.
- The points (1,1) and (-1,1) are on the parabola.
Next, let's draw the parabola 𝑦 = −4(𝑥 + 3)2 − 7:
- It is symmetric about the vertical line x = -3.
- It is shifted downward by 7 units compared to the standard parabola.
- It is vertically compressed by a factor of 4 compared to the standard parabola.
- The vertex is at (-3, -7).
Please note that the sketch will help visualize the transformations and general shape, but it may not be drawn to scale.