To determine the closest graph to the horizontally compressed function \( f(3x) \) from the parent function \( f(x) = x^2 \), we need to understand that compressing the function horizontally by a factor of 3 means that it will become narrower. Specifically, for every \( x \) value, the output will be the same as the parent function, but only at one-third the corresponding \( x \) value.
The original parent function \( f(x) = x^2 \) passes through points such as \( (0, 0) \), \( (1, 1) \), and \( (-1, 1) \). The horizontally compressed function \( f(3x) = (3x)^2 = 9x^2 \) will pass through:
- \( (0, 0) \)
- \( \left(\frac{1}{3}, 1\right) \)
- \( \left(-\frac{1}{3}, 1\right) \)
This means that the graph will be narrower and the x-values of points that would be at \( (1, 1) \) and \( (-1, 1) \) from the original function will instead be at \( \frac{1}{3} \) and \( -\frac{1}{3} \), respectively.
Since we are looking for a graph that represents this effect, we should identify a parabola that is narrow and intersects the y-axis at approximately the same height but with x-axis values stretching closer to the origin compared to the original function.
From the choices given, the most relevant information is that all choices depict an upward parabola. The key detail we are concerned with is the range of the x-axis.
Among the options:
- Ranges from -5 to 5
- Ranges from -10 to 10
- Ranges from -5 to 5 (duplicate of 1)
- Ranges from -5 to 6
Given these ranges, the parabola that corresponds most closely with the notion of compression is:
4. 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.
This choice suggests a more narrowly defined graph that is still capable of displaying the effect of the horizontal compression since the range extends slightly further.
3 answers