Answer :
Certainly! Let's break down the expression [tex]\(y = \frac{1}{3} |x - 4| - 5\)[/tex] step by step.
### Step 1: Understand the Absolute Value
The absolute value function [tex]\(|x - 4|\)[/tex] gives the distance of [tex]\(x\)[/tex] from 4 on the number line, regardless of the direction. It is piecewise defined as:
[tex]\[ |x - 4| = \begin{cases} x - 4 & \text{if } x \geq 4 \\ -(x - 4) & \text{if } x < 4 \end{cases} \][/tex]
### Step 2: Scaling and Shifting
We then scale the absolute value by [tex]\(\frac{1}{3}\)[/tex] and shift the result down by 5 units.
So, the given expression can be rewritten with our understanding of the absolute value function:
[tex]\[ y = \frac{1}{3} |x - 4| - 5 \][/tex]
### Step 3: Piecewise Definition
Using the piecewise definition of the absolute value, we can further break it down into two cases:
1. Case 1: [tex]\( x \geq 4 \)[/tex]:
[tex]\[ |x - 4| = x - 4 \][/tex]
So, substituting [tex]\( |x - 4| \)[/tex] with [tex]\( x - 4 \)[/tex], we get:
[tex]\[ y = \frac{1}{3}(x - 4) - 5 \][/tex]
2. Case 2: [tex]\( x < 4 \)[/tex]:
[tex]\[ |x - 4| = -(x - 4) = 4 - x \][/tex]
So, substituting [tex]\( |x - 4| \)[/tex] with [tex]\( 4 - x \)[/tex], we get:
[tex]\[ y = \frac{1}{3}(4 - x) - 5 \][/tex]
### Step 4: Simplifying Each Case
Let's simplify each case.
#### For [tex]\( x \geq 4 \)[/tex]:
[tex]\[ y = \frac{1}{3}(x - 4) - 5 \][/tex]
Distribute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ y = \frac{1}{3}x - \frac{4}{3} - 5 \][/tex]
Combine the constants:
[tex]\[ y = \frac{1}{3}x - \frac{4}{3} - \frac{15}{3} \][/tex]
[tex]\[ y = \frac{1}{3}x - \frac{19}{3} \][/tex]
#### For [tex]\( x < 4 \)[/tex]:
[tex]\[ y = \frac{1}{3}(4 - x) - 5 \][/tex]
Distribute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ y = \frac{4}{3} - \frac{1}{3}x - 5 \][/tex]
Combine the constants:
[tex]\[ y = \frac{4}{3} - \frac{15}{3} - \frac{1}{3}x \][/tex]
[tex]\[ y = -\frac{11}{3} - \frac{1}{3}x \][/tex]
Or written as:
[tex]\[ y = -\frac{1}{3}x - \frac{11}{3} \][/tex]
### Final Piecewise Function
Combining both cases, the piecewise function for [tex]\( y \)[/tex] is:
[tex]\[ y = \begin{cases} \frac{1}{3}x - \frac{19}{3} & \text{if } x \geq 4 \\ -\frac{1}{3}x - \frac{11}{3} & \text{if } x < 4 \end{cases} \][/tex]
This is the step-by-step breakdown of the given expression [tex]\( y = \frac{1}{3} |x - 4| - 5 \)[/tex].
### Step 1: Understand the Absolute Value
The absolute value function [tex]\(|x - 4|\)[/tex] gives the distance of [tex]\(x\)[/tex] from 4 on the number line, regardless of the direction. It is piecewise defined as:
[tex]\[ |x - 4| = \begin{cases} x - 4 & \text{if } x \geq 4 \\ -(x - 4) & \text{if } x < 4 \end{cases} \][/tex]
### Step 2: Scaling and Shifting
We then scale the absolute value by [tex]\(\frac{1}{3}\)[/tex] and shift the result down by 5 units.
So, the given expression can be rewritten with our understanding of the absolute value function:
[tex]\[ y = \frac{1}{3} |x - 4| - 5 \][/tex]
### Step 3: Piecewise Definition
Using the piecewise definition of the absolute value, we can further break it down into two cases:
1. Case 1: [tex]\( x \geq 4 \)[/tex]:
[tex]\[ |x - 4| = x - 4 \][/tex]
So, substituting [tex]\( |x - 4| \)[/tex] with [tex]\( x - 4 \)[/tex], we get:
[tex]\[ y = \frac{1}{3}(x - 4) - 5 \][/tex]
2. Case 2: [tex]\( x < 4 \)[/tex]:
[tex]\[ |x - 4| = -(x - 4) = 4 - x \][/tex]
So, substituting [tex]\( |x - 4| \)[/tex] with [tex]\( 4 - x \)[/tex], we get:
[tex]\[ y = \frac{1}{3}(4 - x) - 5 \][/tex]
### Step 4: Simplifying Each Case
Let's simplify each case.
#### For [tex]\( x \geq 4 \)[/tex]:
[tex]\[ y = \frac{1}{3}(x - 4) - 5 \][/tex]
Distribute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ y = \frac{1}{3}x - \frac{4}{3} - 5 \][/tex]
Combine the constants:
[tex]\[ y = \frac{1}{3}x - \frac{4}{3} - \frac{15}{3} \][/tex]
[tex]\[ y = \frac{1}{3}x - \frac{19}{3} \][/tex]
#### For [tex]\( x < 4 \)[/tex]:
[tex]\[ y = \frac{1}{3}(4 - x) - 5 \][/tex]
Distribute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ y = \frac{4}{3} - \frac{1}{3}x - 5 \][/tex]
Combine the constants:
[tex]\[ y = \frac{4}{3} - \frac{15}{3} - \frac{1}{3}x \][/tex]
[tex]\[ y = -\frac{11}{3} - \frac{1}{3}x \][/tex]
Or written as:
[tex]\[ y = -\frac{1}{3}x - \frac{11}{3} \][/tex]
### Final Piecewise Function
Combining both cases, the piecewise function for [tex]\( y \)[/tex] is:
[tex]\[ y = \begin{cases} \frac{1}{3}x - \frac{19}{3} & \text{if } x \geq 4 \\ -\frac{1}{3}x - \frac{11}{3} & \text{if } x < 4 \end{cases} \][/tex]
This is the step-by-step breakdown of the given expression [tex]\( y = \frac{1}{3} |x - 4| - 5 \)[/tex].
