Question

1. the function, $f(x) = x^2$ is transformed to create $g(x)$ shown in the graph.

determine which transformations to $f(x)$ created $g(x)$.
first, $f(x)$ was
reflected over the x - axis
reflected over the y - axis
the result was then
translated up 2 units
translated up 4 units

Explanation:

Step 1: Analyze the original function \( f(x) = x^2 \)

The graph of \( f(x) = x^2 \) is a parabola opening upwards with vertex at \((0,0)\).

Step 2: Check reflection over x - axis

If we reflect \( f(x)=x^2 \) over the x - axis, the function becomes \( y=-x^2 \), which is a parabola opening downwards. The graph of \( g(x) \) opens downwards, so reflection over the x - axis is a possible first transformation. Reflecting over the y - axis of \( f(x)=x^2 \) gives \( y = (-x)^2=x^2 \), which is the same as the original function (since \( x^2=(-x)^2 \)) and still opens upwards, so reflection over y - axis is not the first transformation. So the first transformation is reflection over the x - axis.

Step 3: Analyze vertical translation

The vertex of \( y = -x^2 \) is at \((0,0)\). The vertex of \( g(x) \) is at \((0,4)\) (from the graph, since it's symmetric about the y - axis and the peak is at \( y = 4 \)). If we translate \( y=-x^2 \) up 4 units, we get \( y=-x^2 + 4 \), which has a vertex at \((0,4)\) and opens downwards, matching the graph of \( g(x) \). Translating up 2 units would give a vertex at \((0,2)\), which does not match the graph.

Answer:

First, \( f(x) \) was reflected over the x - axis; The result was then translated up 4 units.