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
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.
The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a transformation of the standard parabola 𝑦 = 𝑥^2. Let's break down the transformations one by one:
1. Reflection: The negative sign in front of the equation 𝑦 = -4(𝑥 + 3)2 − 7 indicates a reflection across the x-axis. This means that the parabola opens downwards instead of upwards like the standard parabola.
2. Horizontal Translation: The value inside the parentheses, (𝑥 + 3), represents a horizontal translation of 3 units to the left. This means that the vertex of the parabola is moved 3 units to the left compared to the standard parabola.
3. Vertical Translation: The number outside the parentheses, -7, represents a vertical translation of 7 units downwards. This means that the entire graph of the parabola is shifted 7 units downwards compared to the standard parabola.
Now, let's discuss the steps for drawing a reasonable sketch of both parabolas:
1. Draw the axes: Start by drawing the x and y axes on a coordinate system.
2. Plot the standard parabola 𝑦 = 𝑥^2: Since the standard parabola does not have any translations or reflections, its vertex coincides with the origin (0, 0). Plot a few more points on either side of the vertex to help visualize the shape of the parabola.
3. Plot the transformed parabola 𝑦 = −4(𝑥 + 3)2 − 7: Apply the transformations step by step to plot the transformed parabola.
- Start by translating the vertex of the standard parabola 3 units to the left (since there is a horizontal translation of -3). So the new vertex will be (-3, 0).
- Next, reflect the parabola across the x-axis, resulting in an opening downwards.
- Finally, shift the entire graph 7 units downwards (since there is a vertical translation of -7).
4. Connect the points: Use a smooth, curved line to connect the plotted points on the parabolas, representing the shape of each parabola.
Remember to label the axes and mark the key points, such as the vertex, on the parabolas for a more accurate sketch.