Interpret slope as a rate of change, find the x- and y-intercepts, and graph a line from slope-intercept form (Ohio F-IF.6, A-REI.10).

What this topic is asking

Ohio standards F-IF.6 and A-REI.10 ask you to read slope as a rate of change, find a line's intercepts, and graph it from its equation. Slope and intercepts are the language for describing linear models, so this topic threads through functions and modeling. It spans the Functions and Expressions and Equations reporting categories and appears on both parts.

Slope as a rate of change

Slope measures how steeply a line rises or falls, and in a model it is a rate.

In a context, slope is the output change per one unit of input. For a savings model $B = 150 + 25w$, the slope $25$ means the balance rises $25$ dollars per week. Reading slope as "per one unit" is the link between the number and its meaning.

The intercepts

The two intercepts are where the line crosses the axes, and each is found by setting the other variable to zero.

• $y$-intercept: set $x = 0$. In $y = mx + b$ it is simply $b$, the point $(0, b)$. In context it is the starting value.
• $x$-intercept: set $y = 0$ and solve for $x$. In context it is where the quantity reaches zero (a break-even point or a "runs out" moment).

Graphing from slope-intercept form

Slope-intercept form is built for graphing: it hands you a point and a direction.

On the computer-based test, graphing items have you place points or drag the line; the same "intercept then slope" routine produces the points to use.

How Ohio examines this topic

• Multiple choice. Compute slope from two points or read it from a graph.
• Equation/numeric response. Find an intercept, or a slope, from an equation.
• Graphing. Plot a line by placing its $y$-intercept and a slope-stepped point.

Slope direction and the order of the slope fraction are the usual trap, so keep rise over run with consistent signs.

Why slope is the same everywhere on a line

A defining feature of a straight line is that its slope is constant: pick any two points and the rise-over-run is identical. This is why you can compute slope from whichever two points are convenient and why a line has a single rate of change, unlike a curve. It also justifies the graphing method: once you have one point and the slope, repeatedly stepping "rise then run" lands on more points of the same line, because the steepness never changes. Contrast this with a parabola, whose steepness varies, which is exactly why curves need an average rate of change over an interval rather than a single slope. Recognizing constant slope as the signature of linearity helps you decide, from a table or graph, whether a relationship is linear at all.

Connecting intercepts to a context

Intercepts carry meaning in a modeled situation, and the test likes to ask for it. The $y$-intercept is the value when the input is zero, the starting amount, the flat fee, the initial population. The $x$-intercept is when the output hits zero, the time a balance reaches $0$, the point a quantity is used up. So for a fuel model where $y$ is gallons left after $x$ miles, the $y$-intercept is the tank's starting fuel and the $x$-intercept is the distance at which it runs dry. Tying each intercept to its real-world event is the interpretation step the modeling items reward.

Try this

Q1. Find the slope of the line through $(2, -3)$ and $(6, 5)$. [1 point]

• Cue. $m = \frac{5 - (-3)}{6 - 2} = \frac{8}{4} = 2$.

Q2. Find both intercepts of $y = \frac{1}{2}x - 4$. [2 points]

• Cue. $y$-intercept $(0, -4)$; set $y = 0$: $x = 8$, so $(8, 0)$.