How is the graph of y = |x| − 1 different from the parent function f(x) = |x|?

We can define transformations as operators that act over functions and modify them in some way. One of these transformations is the family of translations, who translate the whole graph of the function in some given direction.

Here we will find that the graph of y = |x| - 1 is a translation of 1 unit downwards of y = |x|.

To see this, we first need to know the vertical translation.

For a given function f(x), a vertical translation of N units is written as:

g(x) = f(x) + N

If N is positive the translation is upwards

if N is negative the translation is downwards,

So if here we have:

f(x) = |x|

then:

g(x) = |x| - 1 = f(x) - 1

So we can see that this is a translation of one unit down. This moves the whole graph of |x| one unit downwards, the graphs can be seen in the image below.

Where the blue graph is the one of y = |x| - 1

If you want to learn more, you can read:

link

Step-by-step explanation:

the graph moves down one unit

if you add/subtract outside the parent function the graph moves up/down

Step-by-step explanation:

Since Segment BOA is a diameter:

Arc Ac and Arc CB are in a ratio of two to four. Since Segment BOA is a diameter, Arc ACB measures 180°. Letting the unknown value be x, we can write that:

Hence:

Thus, Arc CB = 120°. By the Inscribed Angle Theorem:

Therefore, ΔABC is a 30-60-90 triangle. Its sides are in the ratios shown in the image below.

Since AC is opposite from the 30° triangle, let AC = a.

We are given that AC = 9. Hence, a = 9.

BC is opposite from the 60° angle and it is given by a√3. Therefore: