What kind of transformation converts the graph of f(x)=x2+3 into the graph of g(x)=4x2+3?

Answer 1

A horizontal shrink (compression) of 2.

f(2x)²+ 3 ⇒4x² + 3

A vertical stretch would create 4x² + 12.

Note: vertical stretches are applied to the entire function, that is, the output of the function:

a*f(x)

If f(x) = x²+ 3, then 4*f(x) = 4(x²+3) = 4x² + 12.

Horizontal stretches and shrinks are applied to the input of the function, inside the parentheses:

f(ax).

If there was a vertical displacement, as is true in this case, it would be outside the parentheses and would not be affected by a horizontal stretch or shrink.

Answer 2

The transformation that converts the graph of \(f(x) = x^2 + 3\) into the graph of \(g(x) = 4x^2 + 3\) is a vertical stretch (or dilation) by a factor of 4.

In general, when you have a function of the form \(f(x) = a \cdot g(x)\), where \(a\) is a positive constant, it means that the graph of \(f(x)\) has been vertically stretched or compressed compared to the graph of \(g(x)\). In this case, \(a = 4\), which means the graph of \(g(x)\) has been vertically stretched by a factor of 4 compared to the graph of \(f(x)\).

The "+3" part in both functions represents a vertical shift upward by 3 units, but it does not affect the type of transformation (stretching) between the two functions.