Question

A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting

parabola is (-6,3). What is the vertex of the function defined as
g(x) = f(x - 2) – 5?

Answer 1

The vertex of g(x) is (-4, -2)

Explanation:

Let's define general translations:

Vertical translation:

For a function f(x), a vertical translation of N units (So the whole graph of f(x) is translated N units in a given direction) is defined as:

g(x) = f(x) + N

such that:

If N is positive, the translation is upwards

if N is negative, the translation is downwards.

Horizontal translation:

Similar than before, for a function f(x) a horizontal translation of N units is written as:

g(x) = f(x + N)

Where, if N is positive, the translation is to the left

If N is negative, the translation is to the right.

Here, we have the transformation:

g(x) = f(x - 2) - 5

This is a transformation that moves the whole graph of f(x):

2 units to the right

5 units downwards.

So, if a point like (x, y) was in the graph of f(x)

After the translation, that point will be (x + 2, y - 5)

Then, if the original vertex was (-6, 3), the new vertex will be:

(-6 + 2, 3 - 5)

(-4, -2)

The vertex of g(x) is (-4, -2)