## Absolute Value Function Definition

Absolute value functions are important concepts in mathematics, especially when dealing with real numbers. The **absolute value** of a number is its distance from zero on the number line, regardless of direction.

### Understanding Absolute Value

The absolute value of a number, denoted by \(|x|\), is the non-negative value of \(x\), without regard to its sign. This means that \(|x| \geq 0\) for any real number \(x\).

In mathematical terms, the absolute value function can be defined as follows: For any real number \(x\): \[ |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases} \]

Consider the absolute value of different numbers:

- \(|3| = 3\)
- \(|-3| = 3\)
- \(|0| = 0\)

The absolute value function always outputs a positive value or zero, never a negative value.

### Graphing Absolute Value Functions

When graphing absolute value functions, it is essential to note that they create a V-shaped graph that opens upward. The general form of an absolute value function is: \[ f(x) = a|x - h| + k \] Here, \(a\), \(h\), and \(k\) are constants that affect the graph's shape and position.

For instance, consider the function \[ f(x) = 2|x - 1| - 3 \]:

- The graph of \(y = |x| \) shifts 1 unit to the right due to \(x - 1\).
- It is scaled by a factor of 2, causing it to be narrower.
- It moves 3 units down, resulting in a final downward shift due to \(- 3 \).

Absolute value functions are piecewise linear functions. This means they contain linear pieces in the different regions defined by the function's domain. The term functions play a crucial role in real-life applications, including distance measurement and physics. For example, the absolute value can represent the distance error in positioning systems or fluctuations in stock market prices.

## Graphing Absolute Value Functions

When graphing absolute value functions, it is essential to note that they create a V-shaped graph that opens upward. The general form of an absolute value function is: \[ f(x) = a|x - h| + k \] Here, \(a\), \(h\), and \(k\) are constants that affect the graph's shape and position.

For instance, consider the function \[ f(x) = 2|x - 1| - 3 \]:

- The graph of \(y = |x|\) shifts 1 unit to the right due to \(x - 1\).
- It is scaled by a factor of 2, causing it to be narrower.
- It moves 3 units down, resulting in a final downward shift due to \(- 3\).

Absolute value functions are piecewise linear functions. This means they contain linear pieces in the different regions defined by the function's domain. The term functions play a crucial role in real-life applications, including distance measurement and physics. For example, the absolute value can represent the distance error in positioning systems or fluctuations in stock market prices.

Observing the vertex and transformations in the graph of an absolute value function can help you understand its behaviour.

## Absolute Value Function Equation

Absolute value functions are easy to understand once you grasp the basics of absolute value. These functions have a specific equation that determines their graph and behaviour. They generate a unique V-shaped graph and are fundamental in various mathematical applications.

An absolute value function is represented by the equation: \[ f(x) = a|x - h| + k \] Here, \(a\) determines the slope; \(h\) shifts the graph left or right, and \(k\) shifts the graph up or down.

### Components of the Absolute Value Function

**a**: This value affects the graph's steepness or slope. If \(a > 1\), the V shape is narrower. If \(0 < a < 1\), the V shape is wider.**h**: This constant shifts the graph horizontally. If \( h \geq 0 \), move the graph to the right. If \( h < 0 \), move it to the left.**k**: This value shifts the graph vertically. If \( k > 0 \), move the graph upwards. If \( k < 0 \), move it downwards.

Consider the function:\[ f(x) = 3|x + 2| - 5 \]

- The graph of \( y = |x| \) shifts 2 units to the left because of \( x + 2 \).
- The graph is scaled by 3, making it narrower.
- The graph shifts 5 units down due to \( - 5 \).

Try manipulating the values of \(a\), \(h\), and \(k\) yourself to see how the graph changes!

### Graphing an Absolute Value Function

To graph an absolute value function, follow these steps:

- Determine the vertex of the graph, located at \( (h, k) \).
- Plot points on either side of the vertex to define the V shape.
- Connect the points, ensuring the correct slope as determined by the value of \(a\).

Absolute value functions are crucial in fields such as geometry, physics, and economics. For example, in physics, the absolute value can represent the magnitude of force or displacement. In geometry, these functions help calculate the shortest distance between points.

## Absolute Value Function Transformations

Understanding absolute value function transformations can help you grasp more advanced mathematical concepts and solve complex problems. Transformations of the absolute value function affect its shape and position on the graph.

### What is an Absolute Value Function?

An absolute value function is represented by the equation: \[ f(x) = a|x - h| + k \] Here, \(a\) determines the slope; \(h\) shifts the graph left or right, and \(k\) shifts the graph up or down.

- The absolute value of a number \(x\) is its distance from zero on the number line.
- It is always a non-negative value.
- This concept is crucial in measuring distances and other real-world applications.

### Graphing Absolute Value Functions: Steps and Tips

To graph an absolute value function, follow these steps:

- Determine the vertex of the graph, located at \((h, k)\).
- Choose and plot points on either side of the vertex to establish the V shape.
- Connect these points with straight lines, making sure the slope matches the value of \(a\).

Consider the function:\[ f(x) = 2|x - 3| + 1 \]

- The graph of \(y = |x|\) shifts 3 units to the right because of \(x - 3\).
- The graph is scaled by 2, making it narrower.
- The graph shifts 1 unit up due to \(+ 1\).

Remember to use the vertex to guide you when plotting an absolute value function!

### Standard Form of Absolute Value Function Equation

The standard form of an absolute value function equation is: \[ f(x) = a|x - h| + k \] This form is vital in understanding how different values of \(a\), \(h\), and \(k\) transform the graph.

Let's look at a different example:\[ f(x) = -0.5|x + 4| - 2 \]

- The graph of \(y = |x|\) shifts 4 units to the left because of \(x + 4\).
- It is scaled by -0.5, causing it to be wider and flipped over the x-axis.
- It moves 2 units down.

Manipulating the values of \(a\), \(h\), and \(k\) yourself greatly helps in understanding transformations.

### Solving Absolute Value Inequalities

Absolute value inequalities can be solved by considering the definition of absolute value. The key steps involve setting up two conditions and solving both.

For instance, to solve \( |x - 3| < 7 \):

- The inequality can be split into two cases: \( x - 3 < 7 \) and \( -(x - 3) < 7 \).
- Solving these gives \( x < 10 \) and \( x > -4 \).
- So, the solution is \(-4 < x < 10\).

### Real-Life Absolute Value Function Examples

Absolute value functions can be found in various real-life applications, such as calculating distances, financial models, and physics.

An absolute value function could represent the distance a hiker travels from their starting point, regardless of direction.

- For example, if a hiker walks 3 km east (represented by \( +3 \)) and then 4 km west (represented by \( -4 \)), the total distance walked is represented by \( |3| + |-4| = 7 \) km.

In finance, absolute value functions are used in models to capture returns' volatility. They help in understanding how much a stock's price changes, regardless of the direction. This concept aids investors in making informed decisions about their investments.

### Horizontal and Vertical Shifts in Absolute Value Functions

Horizontal and vertical shifts are fundamental transformations that move the graph of an absolute value function without altering its shape.

For example:\[ f(x) = |x - 4| + 5 \]

- The graph of \(y = |x|\) shifts 4 units to the right due to \(x - 4\).
- It shifts 5 units up because of \(+ 5\).

### Reflections and Stretches in Absolute Value Function Transformations

Reflections and stretches are another type of transformation that affect the graph of absolute value functions by changing their orientation and scale.

Consider the function:\[ f(x) = -2|x + 1| \]

- The graph of \(y = |x|\) flips over the x-axis because of the negative sign.
- It is also scaled by a factor of 2, making it narrower.
- The graph shifts 1 unit to the left due to \(x + 1\).

Practising with different values of \(a\), \(h\), and \(k\) can deepen your understanding of these transformations.

## Absolute value functions - Key takeaways

**Absolute value function definition:**The absolute value of a number is its non-negative distance from zero on the number line, denoted by |x|.**Absolute value function equation:**The general form is f(x) = a|x - h| + k, where 'a' affects the slope, 'h' shifts the graph horizontally, and 'k' shifts it vertically.**Graphing absolute value functions:**They create a V-shaped graph that opens upwards. The graph's shape and position are influenced by the constants a, h, and k.**Absolute value inequalities:**To solve, set up two cases based on the definition of absolute value and solve both conditions.**Absolute value function transformations:**Include horizontal and vertical shifts, reflections, and stretches that change the graph's position and orientation.

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