The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a reflection of the standard parabola 𝑦 = 𝑥2 about the x-axis, vertically stretched by a factor of 4, shifted 3 units to the left, and shifted 7 units downward.
To draw a reasonable sketch of both parabolas, we can follow these steps:
1. Plot the vertex of the parabola defined by 𝑦 = 𝑥2 at the origin (0, 0).
2. Plot the vertex of the parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 at (−3, −7), which is 3 units to the left and 7 units downward from the origin.
3. Draw the standard parabola by connecting points on either side of the vertex that are equidistant from it. For example, we can connect (−2, 4), (−1, 1), (0, 0), (1, 1), and (2, 4). We can sketch the curve smoothly, noting that it is symmetric about the y-axis.
4. Draw the transformed parabola by following the same steps as above, but using the transformed equation 𝑦 = −4(𝑥 + 3)2 − 7 instead of 𝑦 = 𝑥2. We can plot points using a table or calculator, or we can use the fact that the vertex, axis of symmetry, and shape are preserved under transformations. Thus, we can shift the standard parabola 3 units to the left, reflect it about the x-axis, and stretch it vertically by a factor of 4. We can connect points on either side of the transformed vertex (−3, −7) that are equidistant from it, such as (−6, −7), (−4, −15), (−3, −19), (−2, −15), and (0, −7). We can sketch the curve smoothly, noting that it is symmetric about the vertical line x = −3.
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, tell the steps of drawing a reasonable sketch of both parabolas
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