Question

Determine the new equation if the graph of y= x² is compressed by a factor of 3 in the y-direction and then shifted 4 units down.

Answer 1

The equation of the graph after compressing the function
`\(y = x^2\)`by a factor of 3 in the y-direction and shifting it 4 units down is
`\(y = (1)/(3)x^2 - 4.\)`

Step-by-step explanation:

To compress the graph in the y-direction by a factor of 3, we need to multiply the entire function by the reciprocal of the compression factor. The general form of a compression or stretch in the y-direction is
`\(y = a \cdot f(x)\)`, where a is the compression or stretch factor. In this case,
`\(a = (1)/(3)\),` so the new equation becomes
`\(y = (1)/(3)x^2\).`

Next, to shift the graph 4 units down, we subtract 4 from the entire function. The general form of a vertical shift is y = f(x) + c where cis the shift amount. Therefore, the final equation after shifting down by 4 units is
`\(y = (1)/(3)x^2 - 4\).`

This means that each y-coordinate of the original function
`\(y = x^2\)`has been compressed by a factor of 3 and then shifted downward by 4 units. The x-coordinate remains unchanged. The resulting graph is narrower and shifted down compared to the original parabola.