# Let g(x) be the reflection of f(x) = x^{2} + 3 on the x-axis. What is the function rule for g(x)?

**Solution: **

Given: Function f(x) = y = x^{2} + 3, g(x) is the reflection of f(x) on the x-axis.

f(x) is a concave up parabola, the vertex is (0,3) and the axis of symmetry is the x-axis (x=0)

If it is reflected in the x-axis (about y=0) then it will still have the y axis as the axis of symmetry but it will be concave down as it is a flip over the x-axis.

To find the function of g(x) we will have to apply the rule of reflection.

The transformation rule for a graph's reflection over the x-axis is (x,y) → (x,−y).

Thus the vertex will be (0, -3)

Applying this rule to the function f(x) we get,

⇒ y = x^{2} + 3

⇒ -y = -(x^{2} + 3)

⇒ -y = - x^{2} - 3

Or we can say that

g(x) = - f(x)

i.e., g(x) = -(x^{2} + 3)

Thus g(x) = - x^{2} -3

## Let g(x) be the reflection of f(x) = x^{2} + 3 on the x-axis. What is the function rule for g(x)?

**Summary: **

The reflection of the function f(x) = x^{2} + 3 across x axis is given by g(x) = -x^{2} - 3.