Answer:




Explanation:
From inspection of the given graph, function f(x) is a parabola that opens downwards. Its vertex is (2, 1/2) and its y-intercept is (0, -1/2).
To determine the equation of f(x), we can use the vertex formula:

Substitute the vertex (2, 1/2) and the y-intercept (0, -1/2) into the formula to determine the value of a:

Substitute the found value of a and the vertex into the formula to create an equation for f(x):


To reflect a function in the y-axis, negate the x-value of each point, but leave the y-value the same:
- Reflection in the y-axis: (x, y) → (-x, y)
Therefore, if g(x) is f(x) reflected in the y-axis, replace x with -x:

The vertex of g(x) is (-2, 1/2) and its y-intercept is (0, -1/2).

To reflect a function in the x-axis, negate the y-value of each point, but leave the x-value the same:
- Reflection in the x-axis: (x, y) → (x, -y)
Therefore, if h(x) is f(x) reflected in the x-axis, then:

The leading coefficient is positive, so the parabola opens upwards.
The vertex of h(x) is (2, -1/2) and its y-intercept is (0, 1/2).

If p(x) is g(x) reflected in the y-axis, then:

The leading coefficient is positive, so the parabola opens upwards.
The vertex of p(x) is (-2, -1/2) and its y-intercept is (0, 1/2).