Graph linear equations in two variables and identify slope and intercepts, labeling axes and scale (NC.M1.A-CED.2, F-IF.4).

What this topic is asking

NC.M1.A-CED.2 asks you to graph equations in two variables on coordinate axes, labeling axes and scales. Paired with NC.M1.F-IF.4 (interpreting key features), this means you both draw a line and read its slope and intercepts. On the online EOC you place points and lines with a graphing tool, so accuracy in plotting matters.

Graphing from slope-intercept form

When the equation is $y = mx + b$, the intercept and slope give you everything.

Graphing from standard form using intercepts

When the equation is $Ax + By = C$, the two intercepts are quickest.

• x-intercept: set $y = 0$ and solve for $x$. The point is $(x, 0)$.
• y-intercept: set $x = 0$ and solve for $y$. The point is $(0, y)$.

Plot both and connect. For $2x + 5y = 10$: x-intercept $(5, 0)$, y-intercept $(0, 2)$.

Reading slope and intercepts from a graph

To read a graph, find the y-intercept where the line crosses the y-axis, then pick two lattice points the line passes through and compute rise over run between them. Counting up/down for rise and right/left for run, with signs, gives the slope. A line falling left to right has a negative slope.

Horizontal and vertical lines

Two special cases appear often:

• Horizontal line $y = c$: slope $0$, crosses the y-axis at $c$, never the x-axis (unless $c = 0$).
• Vertical line $x = c$: slope undefined, crosses the x-axis at $c$, and is not a function (it fails the vertical line test).

How the NC Math 1 EOC examines this topic

• Technology-enhanced. Plot a line by placing points, or drag a line to match an equation.
• Multiple choice. Match a graph to its equation, or identify an intercept or the slope.
• Gridded response. Enter the value of an intercept.

Graphing ties to writing linear equations (the same slope and intercept) and to solving systems by graphing, where the intersection of two lines is the solution.

Why the intercepts and slope are enough

A line has no curvature, so two pieces of information fix it entirely: a starting point and a direction. The y-intercept supplies a point on the y-axis, and the slope supplies the direction; alternatively, the two intercepts supply two points. Either pairing determines the unique line, which is why you never need a table of many points to graph a line accurately. This economy, two facts, one line, is the same idea behind writing equations, and it is what makes linear models the simplest and most common functions on the EOC. Reading a graph is just running the process backward: recover the slope and intercept from the picture.

Try this

Q1. Graph $y = -2x + 3$ by describing the first two points. [1 point]

• Cue. Plot $(0, 3)$; slope $-2$ means down $2$, right $1$ to $(1, 1)$.

Q2. Find both intercepts of $4x - y = 8$. [2 points]

• Cue. x-intercept: $y = 0 \Rightarrow x = 2$, so $(2, 0)$; y-intercept: $x = 0 \Rightarrow y = -8$, so $(0, -8)$.