How to Compare Linear Functions: Equations, Tables, and Graphs

We’ll use three main properties for comparison: the slope (rate of change) and the \(y\)-intercept (starting point), the pattern of change (seen in tables), and the steepness and starting point of lines (seen in graphs).

1. Comparing Linear Functions: Equations vs Tables:

Step 1: From the equation, identify the slope and \(y\)-intercept. In the equation \(y = mx + b\), ‘\(m\)’ is the slope, and ‘\(b\)’ is the \(y\)-intercept.

Step 2: For the table, calculate the rate of change between each pair of points. The change in \(y\) divided by the change in \(x\) gives the rate of change (slope). The \(y\)-value when \(x=0\) is the \(y\)-intercept.

Step 3: Compare the slopes (or rates of change) from the equation and the table. The one with the larger slope increases more quickly.

Step 4: Compare the \(y-intercepts from the equation and the table. The one with the larger \(y-intercept starts higher on the \(y-axis.

2. Comparing Linear Functions: Equations vs Graphs:

Step 1: From the equation, identify the slope and \(y\)-intercept.

Step 2: For the graph, calculate the slope by selecting two points on the graph and calculating the change in \(y\) divided by the change in \(x\). The \(y\)-intercept is the point where the line crosses the \(y\)-axis.

Step 3: Compare the slopes from the equation and the graph. The steeper line has a larger slope.

Step 4: Compare the \(y\)-intercepts from the equation and the graph. The line that crosses the \(y\)-axis at a higher point has a larger \(y\)-intercept.

3. Comparing Linear Functions: Tables vs Graphs:

Step 1: For the table, calculate the rate of change between each pair of points and identify the \(y\)-value when \(x=0\) as the \(y\)-intercept.

Step 2: For the graph, calculate the slope by selecting two points on the graph and calculating the change in \(y\) divided by the change in \(x\). Identify the \(y\)-intercept as the point where the line crosses the \(y\)-axis.

Step 3: Compare the rates of change (or slopes) from the table and the graph. The function represented by a steeper line has a larger rate of change.

Step 4: Compare the \(y\)-intercepts from the table and the graph. The line that crosses the \(y\)-axis at a higher point has a larger \(y\)-intercept.