which is one of the transformations applied to the graph of ( f(x) = x^2 ) to change it into the graph of ( g(x) = 9x^2 - 36x )?
- the graph of ( f(x) = x^2 ) is widened.
- the graph of ( f(x) = x^2 ) is shifted right 4 units.
- the graph of ( f(x) = x^2 ) is shifted down 36 units.
- the graph of ( f(x) = x^2 ) is reflected over the ( x )-axis.

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Accepted Answer

36 units. Now, looking at the options:

- Option 1: Widened. Since a=9 > 1, it's stretched, not widened (widened is a < 1), so wrong.

- Option 2: Shifted right 4 units. h=2, so shifted right 2 units, not 4, so wrong.

- Option 3: Shifted down 36 units. k=-36, so vertical shift down 36 units, which is one of the transformations. So this is correct? Wait, but earlier I thought the vertex was (2, -36), so shifting down 36 units is part of the transformation. Wait, but let's check the options again. The options are:

1. Widened: No.

2. Shifted right 4 units: No.

3. Shifted down 36 units: Yes, because the vertical shift is down 36 units (from k=0 to k=-36).

4. Reflected over x-axis: No, because the coefficient of $ x^2 $ is positive in both f(x) and g(x).

Wait, but earlier when I calculated the vertex, the vertical shift is down 36 units, so option 3 is correct? But wait, let's check with x=0: f(0)=0, g(0)=0. At x=0, both are 0. At x=1: f(1)=1, g(1)=9 - 36=-27. At x=2: f(2)=4, g(2)=36 - 72=-36. So the graph of g(x) at x=2 is -36, which is 36 units below f(2)=4? No, f(2)=4, g(2)=-36, so the difference is 4 - (-36)=40? Wait, no, the vertex of f(x) is (0,0), the vertex of g(x) is (2, -36). So the vertical distance from (0,0) to (2, -36) is 36 units down. So shifting down 36 units is part of the transformation. So option 3 is correct? Wait, but let's check the options again. Wait, maybe I made a mistake in the vertex form. Wait, let's re-express g(x):

$ g(x)=9x^2 - 36x = 9(x^2 - 4x) = 9(x^2 - 4x + 4 - 4) = 9[(x - 2)^2 - 4] = 9(x - 2)^2 - 36 $. So the vertex is (2, -36). So from f(x)=x^2 (vertex (0,0)) to g(x)=9(x - 2)^2 - 36 (vertex (2, -36)), the transformations are: horizontal shift right 2 units, vertical stretch by 9, vertical shift down 36 units. Now, the options are:

- Widened: No, stretched.

- Shifted right 4 units: No, 2 units.

- Shifted down 36 units: Yes, because the vertical shift is down 36 units (from y=0 to y=-36).

- Reflected over x-axis: No, since the coefficient is positive.

So the correct option is the third one: "The graph of $ f(x)=x^2 $ is shifted down 36 units." Wait, but earlier I thought the horizontal shift was 2 units, but the option is about shifting down 36 units, which is part of the transformation. So that's correct.

Wait, but let's confirm with another approach. Let's consider the function $ g(x) = 9x^2 - 36x $. Let's write it as $ g(x) = f(3x) \times 9 - 36 $? No, maybe not. Alternatively, let's plug in x=0: f(0)=0, g(0)=0. x=1: f(1)=1, g(1)=9 - 36=-27. x=2: f(2)=4, g(2)=36 - 72=-36. x=3: f(3)=9, g(3)=81 - 108=-27. x=4: f(4)=16, g(4)=144 - 144=0. So the graph of g(x) is a parabola opening upwards (since coefficient of $ x^2 $ is positive) with vertex at (2, -36), passing through (0,0) and (4,0). The graph of f(x) is a parabola opening upwards with vertex at (0,0), passing through (1,1), (2,4), (3,9), (4,16), etc. So comparing the two graphs, the graph of g(x) is shifted down 36 units (since the vertex is 36 units below the vertex of f(x)) and shifted right 2 units, and stretched vertically. Now, looking at the options, the only one that is a transformation applied is "shifted down 36 units" (option 3) or "shifted right 4 units" (option 2) or "widened" (option 1) or "reflected" (option 4). Since the vertical shift is down 36 units, option 3 is correct.

Wait, but earlier I thought the horizontal shift was 2 units, but the option says 4 units. Maybe there's a mistake in my calculation. Wait, let's check the roots of g(x): g(x)=0 when 9x^2 - 36x=0 → 9x(x - 4)=0 → x=0 or x=4. So the roots are at 0 and 4, so the midpoint is (0 + 4)/2=2, which is the x-coordinate of the vertex, so that's correct. So the horizontal shift from f(x) (which has vertex at 0) to g(x) (vertex at 2) is 2 units right, not 4. So option 2 is wrong. Option 3: shifted down 36 units. The y-coordinate of the vertex of f(x) is 0, and of g(x) is -36, so that's a shift down 36 units. So option 3 is correct.

## Step2: Evaluate each option
- Option 1: Widened. Since the coefficient of $ x^2 $ in $ g(x) $ is 9 (greater than 1), the graph is stretched vertically, not widened (widening occurs when $ 0 < |a| < 1 $). So this is incorrect.

- Option 2: Shifted right 4 units. The vertex of $ f(x) $ is at $ (0,0) $, and the vertex of $ g(x) $ is at $ (2, -36) $, so the horizontal shift is 2 units right, not 4. Thus, this is incorrect.

- Option 3: Shifted down 36 units. The y-coordinate of the vertex of $ f(x) $ is $ 0 $, and the y-coordinate of the vertex of $ g(x) $ is $ -36 $, so the graph is shifted down 36 units. This is a correct transformation.

- Option 4: Reflected over the x-axis. The coefficient of $ x^2 $ in both $ f(x) $ and $ g(x) $ is positive, so there is no reflection over the x-axis (reflection would occur if the coefficient were negative). Thus, this is incorrect.

# Answer:
The correct option is: The graph of $ f(x) = x^2 $ is shifted down 36 units. (The third option)

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