Question
Horizontal Stretches & Compressions Quick Check
1 of 51 of 5 Items
Question
Which graph shows the parent function f(x)=x2
horizontally compressed to form the graph of y=f(3x)
?(1 point)
Responses
An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 5 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
Image with alt text: An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 5 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
An upward parabola passes through 3 plotted points. The x-axis ranges from negative 10 to 10 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
Image with alt text: An upward parabola passes through 3 plotted points. The x-axis ranges from negative 10 to 10 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 6 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
Image with alt text: An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 6 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 5 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
Image with alt text: An upward parabola passes through 3 plotted points. The x-axis ranges from negative 5 to 5 in 1-unit increments and the y-axis ranges from 0 to 10 in 1-unit increments.
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Answers
Answered by
GPT-4o mini
To determine which graph represents the horizontally compressed function \( y = f(3x) \) from the parent function \( f(x) = x^2 \), we need to understand that horizontal compression occurs when we replace \( x \) with a value greater than 1 in the function. The factor in front of \( x \) (in this case, 3) indicates how much the graph will be compressed horizontally.
When \( x \) is multiplied by 3, the points that were previously located at \( (a, f(a)) \) will now be at \( (a/3, f(a)) \). This means that the graph will get "narrower," and its vertex will remain at the same vertical location.
### Characteristics of \( y = f(3x) \):
- The vertex at (0,0) remains the same.
- The graph becomes narrower (stretches vertically) because for the same "y" value, \( x \) will be smaller.
Now, based on the information provided about the graphs:
1. **Graph 1**: Ranges from -5 to 5 (wider)
2. **Graph 2**: Ranges from -10 to 10 (wider)
3. **Graph 3**: Ranges from -5 to 6 (comparable width to Graph 1)
4. **Graph 4**: Same as Graph 1 (wider)
Since we are looking for a graph with a horizontal compression (get narrower), we would expect the x-axis range to be smaller than that of the parent function \( f(x) = x^2 \), which typically has an unrestricted x-axis.
Given that the horizontal compression involves \( y = f(3x) \), the most likely candidate graph will be one that visually appears narrower than the original \( f(x) = x^2 \).
From the options provided:
- Graph 3 (which ranges from -5 to 6) would likely present the best case for a horizontally compressed function relative to the others since it appears to fit better within the typical horizontal reflection of the parabola.
### Conclusion:
The correct answer is likely **Graph 3**, as it shows the necessary characteristics of horizontal compression from the original parent function \( f(x) = x^2 \).
When \( x \) is multiplied by 3, the points that were previously located at \( (a, f(a)) \) will now be at \( (a/3, f(a)) \). This means that the graph will get "narrower," and its vertex will remain at the same vertical location.
### Characteristics of \( y = f(3x) \):
- The vertex at (0,0) remains the same.
- The graph becomes narrower (stretches vertically) because for the same "y" value, \( x \) will be smaller.
Now, based on the information provided about the graphs:
1. **Graph 1**: Ranges from -5 to 5 (wider)
2. **Graph 2**: Ranges from -10 to 10 (wider)
3. **Graph 3**: Ranges from -5 to 6 (comparable width to Graph 1)
4. **Graph 4**: Same as Graph 1 (wider)
Since we are looking for a graph with a horizontal compression (get narrower), we would expect the x-axis range to be smaller than that of the parent function \( f(x) = x^2 \), which typically has an unrestricted x-axis.
Given that the horizontal compression involves \( y = f(3x) \), the most likely candidate graph will be one that visually appears narrower than the original \( f(x) = x^2 \).
From the options provided:
- Graph 3 (which ranges from -5 to 6) would likely present the best case for a horizontally compressed function relative to the others since it appears to fit better within the typical horizontal reflection of the parabola.
### Conclusion:
The correct answer is likely **Graph 3**, as it shows the necessary characteristics of horizontal compression from the original parent function \( f(x) = x^2 \).