Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems (TEKS A.3A).

What this topic is asking

TEKS A.3A is a cornerstone of Reporting Category 2: graph a linear function and read off its key features, the $x$-intercept, $y$-intercept, zeros, and slope. On the redesigned test this is assessed both by traditional questions and by hot-spot graphing, where you click points directly on a grid, so accurate plotting matters as much as recognizing a correct graph.

Reading features from slope-intercept form

Slope-intercept form $y = mx + b$ hands you two features directly: the $y$-intercept is $(0, b)$ and the slope is $m$. To graph, plot $(0, b)$, then apply the slope as rise over run to step to a second point.

For $y = \frac{3}{4}x - 2$: plot $(0, -2)$; the slope $\frac{3}{4}$ means up 3, right 4, to $(4, 1)$; draw the line through both. A zero of the function is the $x$-value where $y = 0$; here $0 = \frac{3}{4}x - 2$ gives $x = \frac{8}{3}$, the $x$-intercept $\left(\frac{8}{3}, 0\right)$.

Reading features from standard form

In standard form $Ax + By = C$, the intercept method is quickest. Set $y = 0$ to find the $x$-intercept; set $x = 0$ to find the $y$-intercept; plot both and connect.

Zeros, intercepts, and the language

STAAR uses precise vocabulary. The $x$-intercept is the point where the graph meets the $x$-axis; the zero of the function is the $x$-value at that point. They describe the same place: the $x$-intercept $(4, 0)$ corresponds to the zero $x = 4$. The $y$-intercept is the point where the graph meets the $y$-axis, the output when the input is 0, which in a context is the initial value.

How STAAR examines this topic

• Multiple choice. Identify an intercept, match an equation to a graph, or read the slope from a graph. Intercept swaps are the standard distractor.
• Hot spot. Plot the line by selecting two points, or click the $x$-intercept directly. Placement must be exact, so use the intercept and slope precisely.
• Inline choice. Choose increasing or decreasing, or the sign of the slope, from a dropdown.

A clarifying idea is that two points determine a line, so the cleanest graphing strategy is always to find two reliable points (the two intercepts, or the $y$-intercept plus one slope step) rather than guessing the steepness.

Horizontal, vertical, and context-restricted lines

Two special cases appear regularly. A horizontal line $y = c$ has slope 0: every output is the same, so it never crosses the $x$-axis (unless $c = 0$) and has no zero. A vertical line $x = c$ has undefined slope: it is not a function, because one input maps to infinitely many outputs, and it has no $y$-intercept unless $c = 0$. Recognizing these prevents the error of trying to write $y = mx + b$ for a vertical line.

In a real-world problem, the key features carry meaning and the graph is often restricted to sensible inputs. For a cost function $C = 15 + 5h$ over hours worked, the $y$-intercept $(0, 15)$ is a fixed charge, the slope 5 is the hourly rate, and there is no $x$-intercept in context because the cost never reaches 0 for $h \ge 0$. STAAR rewards reading a feature and stating what it represents, so pair every intercept or slope you identify with its contextual meaning when the problem provides one.

Try this

Q1. Find both intercepts of $4x - 3y = 12$. [2 points]

• Cue. $x$-intercept: $y = 0 \Rightarrow x = 3$, $(3, 0)$. $y$-intercept: $x = 0 \Rightarrow y = -4$, $(0, -4)$.

Q2. State the slope and $y$-intercept of $y = -2x + 7$. [1 point]

• Cue. Slope $-2$; $y$-intercept $(0, 7)$.