Question

which of the formulas would indicate that the graph h(x) was stretched in the horizontal direction by a factor of 3?​

Answer 1

The correct answer to the question is option (2):
`\( h\left((x)/(3)\right) \).`

To address the question about the horizontal stretching of the graph of a function
`\( f(x) \)` by a factor of 3, we need to understand how transformations affect the graph of a function. Here are the steps to determine the correct transformation:

1. Identify the Base Function: Recognize that the graph in question is of some function
`\( f(x) \)`. We don't need to know the form of \( f(x) \) to understand how it will be transformed.

2. Understand Horizontal Stretching: A horizontal stretch by a factor of
`\( a \)` is achieved by replacing every
`\( x \)` in the function with
`\( (x)/(a) \)`. In this case,
`\( a = 3 \).`

3. Apply the Stretch to the Function: Replace
`\( x \)` with
`\( (x)/(3) \)`in the function
`\( f(x) \).`

4. **Write Down the Transformed Function**: The new function after the horizontal stretch will be
`\( f\left((x)/(3)\right) \).`

5. **Choose the Correct Answer**: Look for the choice that represents the transformation
`\( f\left((x)/(3)\right) \).`

The correct transformation that indicates a horizontal stretch by a factor of 3 is:

`\[ f\left((x)/(3)\right) \]`

This means that for any given value of
`\( x \)` in the original function, its corresponding point on the graph will now be three times further away from the y-axis, thus stretching the graph horizontally. If the original function had a point at
`\( (x, y) \)`, the new function after the stretch will have a corresponding point at
`\( (3x, y) \)`, which means the x-values have been stretched out.

Now, from the options given in the image:

1.
`\( h(3x) \)` represents a horizontal compression by a factor of 3, not a stretch.

2.
`\( h\left((x)/(3)\right) \)` is the correct representation of a horizontal stretch by a factor of 3.

3.
`\( h(x) + 3 \)` represents a vertical shift upwards by 3 units.

4.
`\( 3h(x) \)` represents a vertical stretch by a factor of 3.

Therefore, the correct answer to the question is option (2):
`\( h\left((x)/(3)\right) \).`

Answer 2

2. Normally, to make the function of h(x) increment by 1, you have to increment x by 1. If the function h(x) becomes h(x/3), however, the function increments by 1 as x increments by 3. Thus, to reach the same values vertically, you need to move three times further horizontally.