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Let's break down and solve each question:

### 1. Find the x- and y-intercepts of \( f(x) = 2|x - 5| - 6 \) algebraically.

#### Finding the y-intercept:
To find the y-intercept, set \( x = 0 \):
\[
f(0) = 2|0 - 5| - 6 = 2 \times 5 - 6 = 10 - 6 = 4
\]
So, the y-intercept is at \( (0, 4) \).

#### Finding the x-intercepts:
To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[
2|x - 5| - 6 = 0
\]
First, isolate the absolute value term:
\[
2|x - 5| = 6
\]
\[
|x - 5| = 3
\]
This absolute value equation leads to two cases:
1. \( x - 5 = 3 \)  ⟹  \( x = 8 \)
2. \( x - 5 = -3 \)  ⟹  \( x = 2 \)

So, the x-intercepts are at \( (8, 0) \) and \( (2, 0) \).

### 2. Graph this function on Desmos, then find the domain and range in two different notations.

#### Graph:
The function \( f(x) = 2|x - 5| - 6 \) represents a V-shaped graph opening upwards, with the vertex located at \( (5, -6) \).

#### Domain and Range:
- **Domain**: The function is defined for all real values of \( x \), so the domain is:
  - Interval notation: \( (-\infty, \infty) \)
  - Set notation: \( \{ x \in \mathbb{R} \} \)
  
- **Range**: The lowest point on the graph is the vertex at \( y = -6 \). Since the function opens upwards, \( f(x) \) takes on all values from \(-6\) to \( \infty \):
  - Interval notation: \( [-6, \infty) \)
  - Set notation: \( \{ y \in \mathbb{R} \mid y \geq -6 \} \)

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### Related Questions:
1. How do you determine the vertex of an absolute value function?
2. What transformations are applied to the function \( |x| \) to get \( f(x) = 2|x -

1. Algebraically find the x- and y-intercepts of the graph of f(x) = 2|x - 5| - 6?

2. Graph this function on Desmos then find the domain and range of the graph of f(x) in two different notations.

Learn how to algebraically find the x- and y-intercepts, and determine the domain and range of the absolute value function f(x) = 2|x - 5| - 6. This guide covers intercepts, graphing techniques, and notation for domain and range. Suitable for high school students in Grades 10-12.

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