Lesson 5: Analyzing and Sketching Graphs of Functions

Property

The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function.

Examples

• The function $y = 2x + 5$ is increasing because the slope $m=2$ is positive.
• The function $y = -3x + 1$ is decreasing because the slope $m=-3$ is negative.
• The function $y = 9$ is constant because the slope $m=0$. Its graph is a horizontal line.

Explanation

The sign of the slope tells you the direction of the line. A positive slope means the line goes uphill from left to right. A negative slope means it goes downhill. A zero slope means the line is perfectly flat.

Property

Graphed data can reveal trends, suggest relationships, or uncover anomalies. By analyzing the graph of a function, we can describe its behavior qualitatively.
An increasing function is one where the graph rises from left to right, meaning the $y$-value increases as the $x$-value increases.
A decreasing function is one where the graph falls from left to right.
Sudden changes or deviations from an expected pattern, or anomalies, can indicate specific events, such as a thunderstorm affecting temperature data.

Examples

• A graph of a bathtub's water level shows the level increasing steadily as it fills, then sharply rising as a person gets in, staying constant, and finally decreasing as it drains. This models the entire process.
• The graph of a bouncing ball's height over time starts at a peak, drops to zero, and repeatedly bounces back to progressively lower heights. The curve shows the loss of energy with each bounce.
• A graph showing the number of customers in a store over a day would be low in the morning, rise to a peak around lunchtime, dip in the afternoon, and then rise again before closing.

Explanation

The shape of a graph tells a story. Where it goes up, down, flattens out, or suddenly jumps reveals how one variable is affected by another. Analyzing these features helps us understand the real-world situation being modeled.