Skip to main content
TexasMathsSyllabus dot point

How do you find the slope and rate of change of a linear function from a table, a graph, two points, or a context, and interpret it?

Calculate the rate of change (slope) of a linear function represented tabularly, graphically, or algebraically, and interpret slope and intercepts as rate and initial value in context (TEKS A.3A, A.3B).

A STAAR Algebra I answer on finding slope and rate of change from tables, graphs, two points, and contexts (TEKS A.3A, A.3B), the slope formula on the reference sheet, and interpreting slope and intercepts in real-world situations.

Generated by Claude Opus 4.810 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section
  1. What this topic is asking
  2. Slope from two points
  3. Slope from a table
  4. Slope from a graph
  5. Interpreting slope and intercept in context
  6. How STAAR examines slope
  7. Try this

What this topic is asking

Slope is the heartbeat of the two linear reporting categories, which together are over half the STAAR Algebra I test. TEKS A.3A asks you to calculate the rate of change (slope) of a linear function from a table, a graph, or an equation, and A.3B asks you to interpret slope and intercepts in a real-world context. The slope formula is on the reference sheet, so the points come from applying it cleanly and from explaining what the number means.

Slope from two points

The reference-sheet formula subtracts the yy-coordinates over the xx-coordinates, in the same order for both.

m=y2−y1x2−x1.m = \frac{y_2 - y_1}{x_2 - x_1}.

For (1,4)(1, 4) and (5,16)(5, 16): m=16−45−1=124=3m = \frac{16 - 4}{5 - 1} = \frac{12}{4} = 3. The single most common error is reversing the order in only one of the differences, which flips the sign. A safe rule: whichever point you call "point 2", use its coordinates first in both the numerator and the denominator.

Slope from a table

A table represents a linear function when the output changes by a constant amount for equal steps in the input. Divide the change in yy by the change in xx between any two rows.

xx 0 2 4 6
yy 5 11 17 23

From x=0x = 0 to x=2x = 2, yy rises by 6, so m=62=3m = \frac{6}{2} = 3. The constant difference confirms it is linear, and the value at x=0x = 0 (here 5) is the yy-intercept.

Slope from a graph

On a graph, slope is rise over run: pick two lattice points the line passes through exactly, count the vertical change (rise) and the horizontal change (run), and divide. A line going up to the right has positive slope; down to the right, negative; a horizontal line has slope 0; a vertical line has undefined slope.

Interpreting slope and intercept in context

This is where A.3B earns its points and where many students stop short.

A full interpretation always names the quantity, the direction, and the units: "the cost rises by 10 cents per minute", not just "the slope is 0.1".

How STAAR examines slope

  • Multiple choice. Compute slope from two points or a table, with sign-error distractors.
  • Inline choice. Choose both the rate (with sign) and its meaning (filling or draining, increasing or decreasing) from dropdowns.
  • Equation editor and graphing. Enter a rate, or use slope to place a second point on a line.

A clarifying idea is that "constant rate of change" is what makes a relationship linear in the first place: if the table's yy-difference is not constant for equal xx-steps, the relationship is not linear, and a single slope does not describe it.

Try this

Q1. Find the slope through (3,−2)(3, -2) and (7,10)(7, 10). [1 point]

  • Cue. m=10−(−2)7−3=124=3m = \frac{10 - (-2)}{7 - 3} = \frac{12}{4} = 3.

Q2. A taxi charges C=3+2dC = 3 + 2d for dd miles. Interpret the 2. [2 points]

  • Cue. The fare rises by $2 per mile travelled.

Exam-style practice questions

Practice questions written in the style of TEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

STAAR (style)1 marksMultiple choice. A line passes through the points (−2,3)(-2, 3) and (4,−9)(4, -9). What is the slope of the line? (A) −2-2 (B) 22 (C) −12-\dfrac{1}{2} (D) −13-\dfrac{1}{3}
Show worked answer →

The correct answer is (A).

The reference sheet gives slope m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Using (−2,3)(-2, 3) and (4,−9)(4, -9): m=−9−34−(−2)=−126=−2m = \frac{-9 - 3}{4 - (-2)} = \frac{-12}{6} = -2. The most common error is mixing the order of subtraction between the numerator and the denominator, for example computing −9−3−2−4\frac{-9 - 3}{-2 - 4}, which gives the wrong sign. Keep the same point's coordinates first in both differences.

STAAR (style)2 marksInline choice. A pool drains at a constant rate. After 2 minutes it holds 480 gallons; after 7 minutes it holds 330 gallons. The rate of change is [30 / -30 / 150 / -150] gallons per minute, which means the pool is [filling / draining].
Show worked answer →

The rate of change is −30-30 gallons per minute, and the pool is draining.

Treat (minutes, gallons) as points: (2,480)(2, 480) and (7,330)(7, 330). Slope =330−4807−2=−1505=−30= \frac{330 - 480}{7 - 2} = \frac{-150}{5} = -30 gallons per minute. The negative sign means the volume decreases over time, so the pool is draining. The magnitude 30 is the rate; the sign carries the direction. Choosing 150150 ignores the division by the 5-minute change, a frequent slip.

Related dot points

Sources & how we know this