Which graph shows the parent function f(x)=x^2

Question

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 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.

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 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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Answer 1

To determine which graph represents the function \( y = f(3x) \) where \( f(x) = x^2 \), we need to understand how this transformation affects the graph of the parent function.

The function \( y = f(3x) \) indicates that the horizontal scaling of the parent function \( f(x) = x^2 \) is being compressed by a factor of 3. This means that the new graph will be narrower than the original parabola \( y = x^2 \).

1. The parent function \( y = x^2 \) has its vertex at the origin (0,0) and passes through points like (1,1), (2,4), (-1,1), and (-2,4).
2. The transformed function \( y = f(3x) = (3x)^2 = 9x^2 \) will also have its vertex at the origin and will pass through points that have been horizontally compressed by a factor of 3. For instance:
   • The point (1/3, 1) (since \( 3 \times \frac{1}{3} = 1 \))
   • The point (2/3, 4) (since \( 3 \times \frac{2}{3} = 2 \))
   • It will still pass through the y-axis at (0,0).

With this in mind, the x-axis range should be narrower than the original function's range (which significantly includes points from at least -2 to 2 on the x-axis).

Since you mention that the x-axis ranges in the provided options:

• If the graph's x-values are between -5 and 5 or even between -10 and 10, those would be too wide to show the compressed function properly.

Considering all this, the correct graph will be narrower. Typically, you would expect to see changes in x-values up to about -1.5 to 1.5 after compression, so a range of -2 to 2 would show the compressed graph accurately.

Out of the given options, select the graph that shows an upward parabola that fits this description with narrower x-axis values.

Look for the graph that indicates the x-values compressed; the options show:

• An upward parabola passing through 3 points, x-axis usually ranging from -5 to 5 should match this compression.

If you have two similar options, ensure one is indeed narrower if one states a range of x-values from -5 to 5.

Thus, review the final details of the options to select the most appropriate answer confirming the compression.