STAAR Algebra I: a complete guide to describing and graphing linear functions, equations, and inequalities

What this category demands

The Describing and Graphing Linear Functions, Equations, and Inequalities reporting category (TEKS A.2, A.3, A.4) is about 26 percent of the STAAR Algebra I test, and together with the writing-and-solving category it is more than half the points. The skills are: compute and interpret slope, graph lines and read their features, write equations in all three forms, handle parallel and perpendicular lines and direct variation, graph two-variable inequalities, and analyze scatterplots. This guide ties together the dot-point pages, each with its own practice: slope and rate of change, graphing linear functions and key features, writing equations of lines, parallel and perpendicular lines and direct variation, graphing linear inequalities, and scatterplots, trend lines, and correlation.

Slope and key features

Slope is the constant rate of change, $m = \frac{y_2 - y_1}{x_2 - x_1}$, read as change in $y$ per unit change in $x$. From a table it is the constant output difference over the input difference; from a graph it is rise over run. In a context, slope is a rate with units and the $y$-intercept is the initial value. A line's key features are its slope, $y$-intercept (set $x = 0$), and $x$-intercept or zero (set $y = 0$). From $y = mx + b$ read $b$ and $m$ directly; from $Ax + By = C$ use the intercept method.

Writing equations

All three forms are on the reference sheet. Use slope-intercept $y = mx + b$ when you know the slope and $y$-intercept; point-slope $y - y_1 = m(x - x_1)$ when you know the slope and any point; from two points, find the slope first, then use point-slope. Convert to standard form $Ax + By = C$ by clearing fractions and making $A$ a non-negative integer.

Special lines and inequalities

Parallel lines share a slope; perpendicular lines have negative-reciprocal slopes (flip and negate). Direct variation is $y = kx$, a line through the origin with $k = \frac{y}{x}$, not on the reference sheet. For a two-variable inequality, draw the boundary line (solid for $\le$ or $\ge$, dashed for $<$ or $>$), then shade the solution half-plane, using a test point such as $(0, 0)$.

Scatterplots and correlation

A trend line follows the overall pattern of a scatterplot. Correlation has a direction (positive, negative, or none) and a strength (strong or weak), summarized by the correlation coefficient $r$ between $-1$ and $1$. Use the trend-line equation to predict, then judge reasonableness, remembering that correlation is not causation.

How this category is examined

• Multiple choice. Compute slope, identify intercepts, match an equation to a graph, classify correlation, or pick a parallel or perpendicular line.
• Hot spot. Plot a line by selecting points, or select the inequality matching a shaded region. Placement and line style must be exact.
• Equation editor and number entry. Build a line's equation or a trend line and compute a prediction; solve a direct-variation problem.
• Inline choice. Choose increasing or decreasing, the sign of a slope, or the type of correlation.

Check your knowledge

Work these as you would for credit on the redesigned test.

1. Find the slope through $(2, -1)$ and $(6, 11)$. (1 point)
2. State both intercepts of $5x + 2y = 20$. (2 points)
3. Write the line through $(0, -4)$ with slope $\frac{1}{2}$. (1 point)
4. Write the line through $(1, 2)$ and $(4, 11)$ in slope-intercept form. (2 points)
5. Write the line perpendicular to $y = 4x - 1$ through $(0, 3)$. (2 points)
6. $y$ varies directly with $x$; $y = 24$ when $x = 6$. Find $y$ when $x = 9$. (2 points)
7. State the boundary style and shading for $y \le -2x + 5$. (1 point)
8. A trend line passes through $(0, 20)$ and $(5, 45)$. Write its equation and predict $y$ at $x = 12$. (2 points)