Question

Plot the graph of \(y=|3 x|\).

Short answer

Graph the lines \( y = 3x \) for \( x \geq 0 \) and \( y = -3x \) for \( x < 0 \).

Step-by-step solution

- Understand Absolute Value Function

The absolute value function, denoted as \(|x|\), outputs the positive magnitude of a number. It turns negative values into positive values while leaving positive values unchanged.

- Simplify the Given Function

The given function is \( y = |3x| \). For this function, any input value of \( x \) will be multiplied by 3 first, and then the absolute value will be taken.

- Break Down the Function into Cases

To graph \( y = |3x| \), consider the definition of absolute value and break it into two cases: \((3x \geq 0) \) and \((3x < 0) \). This translates to: \( y = 3x \) for \( x \geq 0 \, \) and \( y = -3x \) for \( x < 0 \).

- Plot the Case for \( x \geq 0

For \( x \geq 0 \, \), calculate values of \( y = 3x \, \) and plot the points. Examples include \( (0,0), (1,3), (2,6) \).

- Plot the Case for x < 0

For \( x < 0 \), calculate values of \(( y = -3x \), and plot the points. Examples include \( (0,0), (-1,3), (-2,6) \).

- Draw the Graph

Combine the points from both cases and draw lines through them. This forms a 'V' shape with the vertex at \( (0,0) \) and extending upwards.

Key concepts

absolute value

The concept of absolute value is very important in mathematics, especially when graphing functions. Absolute value, denoted as \(|x|\), refers to the distance a number is from zero on a number line, without considering its direction. For example, both -3 and 3 have an absolute value of 3, because both are three units away from zero. In mathematical terms:
\[ |x| = \left \{ \begin{array}{ll} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{array} \right. \]
This simple yet powerful concept allows us to manage and visualize relationships in different scenarios by transforming negative values into positive values.

piecewise functions

A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the overall domain. In our example, the function \(|3x|\) can be broken down into two cases based on the value of x:

• For \(x \geq 0\), \(y = 3x\)
• For \(x < 0\), \(y = -3x\)

Notice how each piece of the function is defined based on whether x is non-negative or negative. This ensures the function behaves differently in different sections of the domain, which is crucial for accurately plotting the graph.

graph plotting

Graph plotting involves carefully drawing a function on the coordinate plane to visualize its behavior. For \(y = |3x|\), we need to plot points from our piecewise-defined function:

• For \(x \geq 0\), plot points like \( (0,0) \), \((1,3)\), \((2,6)\)
• For \(x < 0\), plot points like \((0,0)\), \((-1,3)\), \((-2,6)\)

By marking these points on the graph and drawing straight lines between them, you'll notice the resulting shape is a V. The vertex of this V is at the origin \((0,0)\), and the arms extend upwards on both sides.

algebra

Understanding algebra is essential when working with functions and their graphs. Here, algebraic manipulation allows us to break down the function \(y = |3x|\) into simpler parts. Recognizing that absolute value changes the sign of negative inputs helps in defining the function piecewise. Additionally, algebra helps in solving for different values of x and y to accurately plot the points on the graph. Mastering these algebraic techniques simplifies many complex mathematical problems.