91Ó°ÊÓ

Graphing linear equations is a fundamental skill in algebra. A linear equation is a math statement that describes a straight line when graphed on a coordinate plane. The general form is given by the equation: \[ y = mx + b \] where:

• \( m \) is the slope, representing the tilt or steepness of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To graph a linear equation:
1. Identify the slope and y-intercept from the equation.
2. Plot the y-intercept on the y-axis.
3. Use the slope to find another point. For example, if the slope \( m \) is 2, move up 2 units and 1 unit to the right from the y-intercept.
4. Draw a line through these points.
Combining multiple linear equations in a piecewise function can create a graph with more complex shapes, such as those with distinct segments.

A function is a relation that uniquely assigns an output to each input. The notation \( f(x) \) is often used to represent functions, with \( x \) as the input and \( f(x) \) as the output. A function can be depicted graphically, showing the relationship between \( x \) and its corresponding \( f(x) \).
This exercise is about understanding a piecewise linear function. A piecewise function is defined by different expressions for different parts of its domain. For instance, the function\( f(x) \) described by:
\[ f(x) = \begin {cases} 4 - x & \text{if } x < 2 \ 1 + 2x & \text{if } x \geq 2 \end {cases} \]
means that:

• For \( x < 2 \), the function uses the expression \( 4 - x \).
• For \( x \geq 2 \), the function switches to the expression \( 1 + 2x \).

This type of function can show breaks or jumps in its graph due to the different expressions used in different intervals.

Understanding slope and y-intercept is crucial when working with linear equations. The slope \( m \) describes how steep a line is. It is calculated as the rise over run (\( \Delta y/ \Delta x \)) which means how much the y-coordinate changes per unit change in the x-coordinate.

• Positive slopes go upwards, and negative slopes go downwards.
• The y-intercept \( b \) indicates where the line crosses the y-axis. It is the value of \( y \) when \( x = 0 \).

Consider the pieces in the given function:
- For \( 4 - x \), the slope \( m = -1 \) and the y-intercept \( b = 4 \). This line decreases as \( x \) increases.
- For \( 1 + 2x \), the slope \( m = 2 \) and the y-intercept \( b = 1 \). This line increases as \( x \) increases.
When graphing these, smoothly connect your points to form the desired segments of the piecewise function. Always consider the appropriate intervals for each piece.