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 draw a reasonable sketch of both parabolas
The parabola defined by 𝑦 = −4(𝑥 + 3)2 − 7 is a vertical compression and reflection of the standard parabola.
Transformations:
1. Reflection across the x-axis (negative coefficient of 𝑦)
2. Horizontal translation 3 units to the left (opposite of the sign in the parentheses)
3. Vertical compression by a factor of 4 (coefficient in front of the squared term)
4. Vertical translation 7 units downward
Steps to draw a reasonable sketch of both parabolas:
1. Plot the vertex of 𝑦 = 𝑥2 at (0, 0)
2. Plot the vertex of 𝑦 = −4(𝑥 + 3)2 − 7, which is shifted 3 units to the left and 7 units downward from the vertex of 𝑦 = 𝑥2, at (-3, -7)
3. Sketch the vertical compression of the parabola by making the arms of the parabola more narrow than the standard parabola
4. Sketch the reflection across the x-axis by making the parabola open downward instead of upward
5. Connect the curve smoothly and label both parabolas.
Here is an example sketch:
BZZZZT! But thanks for playing.
3. Vertical dilation by a factor of 4
My apologies, you are correct. The correct transformation should be vertical dilation by a factor of 4, not compression. Thank you for bringing it to my attention!
To understand how the parabola defined by 𝑦 = −4(𝑥 + 3)² − 7 compares to the standard parabola 𝑦 = 𝑥², we need to identify and describe the transformations applied to the standard parabola that result in the given equation.
1. Horizontal Translation: The term (𝑥 + 3) inside the function means that the graph is shifted horizontally 3 units to the left compared to the standard parabola.
2. Vertical Reflection: The negative sign (-4) in front of the function causes a vertical reflection of the graph. This means the parabola will open downwards instead of upwards.
3. Vertical Stretch: The coefficient -4 stretches the graph vertically, making it narrower compared to the standard parabola.
4. Vertical Translation: The term -7 at the end of the equation shifts the graph vertically 7 units downwards.
To draw a reasonable sketch of both parabolas, follow these steps:
Step 1: Draw the Standard Parabola
- Draw a coordinate axis.
- Mark the x-intercepts at (1, 0) and (-1, 0).
- Mark the vertex at (0, 0).
- Draw the parabolic curve based on these points, opening upwards.
Step 2: Apply Transformations to the Standard Parabola
- Based on the transformations described earlier, shift the entire graph 3 units left.
- Vertically reflect the graph.
- Stretch the graph vertically.
- Shift the graph 7 units downwards.
Step 3: Draw the Transformed Parabola
- Mark the x-intercepts of the transformed parabola, if applicable.
- Mark the vertex of the transformed parabola.
- Draw the transformed parabolic curve according to these points.
By following these steps, you can easily draw a reasonable sketch of both the standard parabola and the transformed parabola 𝑦 = −4(𝑥 + 3)² − 7.