91Ó°ÊÓ

The absolute value function, denoted as \( |x| \), is a fundamental concept in algebra that helps in measuring the distance of a number from zero on the number line. The function defines a 'V' shaped graph that has its vertex at the origin (0,0). Here's why it's important: The absolute value of any number is always non-negative because it represents a distance. Therefore, \( |x| \) will always return zero or a positive number.

When analyzing a function like \( f(x) = 3|x| \), it's crucial to understand that the absolute value function affects the entire behavior of the equation. The graph of \( |x| \) starts at the origin, extends to the right into positive values, and to the left into negative values. Each point on this graph represents the absolute value of corresponding x-values.

The domain and range are vital concepts for understanding the scope and limits of a function. The **domain** of a function refers to the set of all possible input values (x-values). For the function \( f(x) = 3|x| \), the domain encompasses all real numbers because an absolute value function can accept any real number as an input.

In interval notation, this is expressed as \( (-\rightarrow, \rightarrow) \), meaning from negative infinity to positive infinity. The **range** of a function represents all possible output values (y-values). For \( f(x) = 3|x| \), the range extends over non-negative real numbers because the absolute value function yields only zero or positive results, and this specific equation multiplies those outputs by 3.

Therefore, the range is \( [0, \rightarrow) \), indicating all values from 0 upwards to positive infinity.

A vertical stretch is a transformation applied to the graph of a function that scales it vertically by a certain factor. For the function \( f(x) = 3|x| \), the factor is 3. This means every point on the basic graph of \( |x| \) is stretched vertically three times its original distance from the x-axis.

In practical terms, take a point like (1,1) on the graph of \( |x| \). When the vertical stretch by a factor of 3 is applied, this point shifts to (1, 3). Essentially, you multiply the y-coordinate of each point by 3:

• The point (1,1) becomes (1,3)
• The point (-1,1) becomes (-1,3)
• The point (2,2) becomes (2,6)

This stretching maintains the 'V' shape but makes it narrower as it pulls the arms of the graph away from the x-axis more steeply, considering all points are stretched thrice as high.

Hence, each y-value on the graph of basic \( |x| \) is increased by the factor of 3.