Find the slope and intercepts of a linear function and interpret them in context, working from an equation, a graph, or a table (LA A1: A-REI.D, F-IF.B).

What this topic is asking

This dot point asks you to find and interpret slope and intercepts of a linear function from an equation, a graph, or a table, drawing on A-REI.D (graphing) and F-IF.B (key features). On LEAP 2025 these are Type I Major Content items, and interpreting slope as a rate is a frequent Type II reasoning prompt. Note that the slope formula and the line forms are not on the reference sheet.

Finding slope

From two points, use the slope formula. From a graph, count rise over run between two lattice points. From a table, divide the change in $y$ by the change in $x$ between two rows.

A table is linear exactly when the slope between every pair of rows is the same.

Slope-intercept form and the intercepts

In $y = mx + b$, you can read the slope and $y$-intercept directly. The $y$-intercept is where the line crosses the $y$-axis (set $x = 0$); the $x$-intercept is where it crosses the $x$-axis (set $y = 0$ and solve).

Interpreting slope and intercept in context

When a line models a real situation, the slope is a rate of change and the $y$-intercept is a starting value. For $y = 5x + 30$ (savings after $x$ weeks), the slope $5$ is "dollars saved per week" and the intercept $30$ is "starting savings." Always state the units.

How LEAP examines this topic

• Equation response. Compute a slope or an intercept and enter the value.
• Type II reasoning. Interpret the slope or intercept of a model in words, with units.
• Graphing item. Identify slope and intercepts from a graph, or plot a line from them.

Why slope captures a constant rate

A line is exactly the graph of a relationship with a constant rate of change, which is the conceptual reason slope is the single number that defines its steepness. Between any two points on the same line, the ratio $\frac{\text{rise}}{\text{run}}$ is the same, because the line climbs (or falls) by the same amount for every step right. That constancy is what makes a relationship linear in the first place: a savings plan that adds $5 every week, a taxi that charges$2 per mile, a tank that drains at a fixed rate. The $y$-intercept then fixes where the line sits, the value before any change has happened. Together, slope and intercept pin down the whole line, which is why slope-intercept form $y = mx + b$ is so useful: $m$ answers "how fast?" and $b$ answers "starting from where?" Recognizing slope as a rate also connects this topic to average rate of change, where the same idea is applied to functions that are not lines.

Try this

Q1. Find the slope through $(-2, 5)$ and $(2, -3)$. [2 points]

• Cue. $m = \dfrac{-3 - 5}{2 - (-2)} = \dfrac{-8}{4} = -2$.

Q2. Find the $x$-intercept of $y = 3x + 12$. [2 points]

• Cue. Set $y = 0$: $0 = 3x + 12 \Rightarrow x = -4$.