Interpretation of linear functions

1. What is a Linear Function?

A linear function represents a straight line when graphed. It can be written in the general form: \[ y = mx + b \]

Here:

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

2. The Slope (\( m \)):

The slope is a measure of the steepness or the inclination of the line. It represents the rate of change of one quantity with respect to another.

• If \( m > 0 \), the function is increasing: as \( x \) increases, \( y \) increases.
• If \( m < 0 \), the function is decreasing: as \( x \) increases, \( y \) decreases.
• If \( m = 0 \), the function is constant: \( y \) doesn't change as \( x \) changes.

The value of \( m \) can be interpreted as "rise over run." That is, for every increase of 1 unit in \( x \), \( y \) will increase by \( m \) units.

3. The Y-Intercept (\( b \)):

The y-intercept, \( b \), is the value of \( y \) when \( x = 0 \). It tells us where the line intersects or "cuts" the y-axis. In real-world scenarios, this can often represent a starting value or a base amount before any changes are considered.

4. Real-World Interpretation:

Linear functions often model real-world scenarios where there's a consistent rate of change.

Example: Suppose you're saving money and you decide to save $50 every week. You already have $200 saved up.

• Equation: The linear function that represents this scenario is \( y = 50x + 200 \).
• Interpretation:
  • The slope \( m = 50 \) means that for every week (increase by 1 in \( x \)), your savings increase by $50.
  • The y-intercept \( b = 200 \) tells us that you started with $200.

If you want to know how much you'll have saved up after 10 weeks, plug in \( x = 10 \): \[ y = 50(10) + 200 = 700 \] After 10 weeks, you'll have $700.

5. Graphical Interpretation:

When graphing the function \( y = 50x + 200 \):

• Start by plotting the y-intercept (0, 200).
• From there, use the slope to find the next point. Since the slope is 50 (or 50/1), you'll rise 50 units and run 1 unit to the right.
• Continue plotting points and then draw a straight line through them.

Final Thoughts:

Linear functions provide a straightforward way to understand and predict relationships that have a constant rate of change. They are foundational in mathematics and are commonly used in various real-world applications, from finance to physics. Understanding the components of a linear function, especially the slope and y-intercept, provides valuable insights into the nature of the relationship it represents.