Question

What is the vertex of the graph g(x)= |x-9| - 3 ?

a) (-9,3)
b) (-3,-9)
c) (9,-3)
d) (9,3)

Answer 1

You have to do this type of question in steps.
g(x) = |x-9| - 3 can be written without the absolute value signs if you split the graph into 2 sections.

Here's an easier example. For the function y = |x| when x < 0 (negative values for x), the function becomes y = -x eg. x = |-5| therefore x = -(-5) = 5
When x>=0, the function becomes y = x.
The absolute value signs change something negative into a positive. They don't do anything to a positive number.

First you look at where x-9 = 0 That happens at x=9 so we're going to look numbers smaller than 9 and then numbers larger than 9. Those are out two sections (intervals).

If x < 9 the stuff in the absolute value sign is negative, so we write out the function as g(x) = -(x-9) - 3, for x < 9 so g(x) = -x + 6, for x < 9
If x>=9 the stuff in the absolute value sign is positive so we can simply remove the absolute value signs.
g(x) = x-9-3, for x >=9 so g(x) = x - 12, for x >=9
If you graph these lines, you'll see that it's a negative slope and then at x = 9 it changes to be a positive slope. At x = 9, g(x) = -3 That's where the vertex is.
The answer is c) (9,-3)