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GED Math: Graphs, Slope, and Functions Concepts

Importance ★★★★★ · 30% (estimate)

Overview

This chapter covers the coordinate plane, slope, equations of lines, parallel and perpendicular lines, function notation, and reading key features of graphs and tables. Many of the algebra targets in the GED assessment guide involve graphs and functions.

A point (x, y) is plotted by moving x units left or right and then y units up or down from the origin. The four quadrants are numbered counterclockwise starting from the upper right. Slope measures steepness: rise over run, or the change in y divided by the change in x. A line going up from left to right has a positive slope.

In slope-intercept form, y = mx + b, m is the slope and b is the y-intercept, where the line crosses the y-axis. In real situations, the slope is a rate such as dollars per hour, and the y-intercept is a starting amount such as a flat fee. The point-slope form, y − y₁ = m(x − x₁), writes the equation of a line from one point and a slope. Parallel lines have equal slopes, and perpendicular slopes are negative reciprocals.

A function gives exactly one output for each input. Function notation f(5) means put 5 in for x. Key features of a graph include its intercepts, where it rises or falls, and its highest or lowest point. To compare two functions shown in different ways, find each one's rate of change and starting value.

Key concepts

  • Slope

    m = (y₂ − y₁) / (x₂ − x₁), the rise divided by the run. Positive slopes go up to the right, negative slopes go down.

    Memory tipSubtract the y-values and x-values in the same order.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Slope-intercept form

    y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0).

    Memory tipIn a cost formula, m is the rate and b is the starting fee.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Point-slope form

    y − y₁ = m(x − x₁) gives the line with slope m through the point (x₁, y₁).

    Memory tipDistribute and solve for y to change it to y = mx + b.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

  • Parallel and perpendicular

    Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals, such as 2/3 and −3/2, which multiply to −1.

    Memory tipFlip the fraction and change the sign.

  • Function

    A rule that gives exactly one output for each input. A table or set of pairs is not a function if one x-value has two different y-values.

    Memory tipTwo inputs sharing the same output is allowed.

  • Key features

    x-intercepts (where y = 0), y-intercept (where x = 0), intervals where the graph increases or decreases, and maximum or minimum points.

    Memory tipFor a table, the maximum is the largest output value.

See it in a picture

Coordinate grid from −6 to 6 with the line y = (2/3)x + 1. The point (0, 1) on the y-axis is labeled as the y-intercept. A dashed path goes from (0, 1) 3 units right, labeled run = 3, then 2 units up to (3, 3), labeled rise = 2. A note reads slope = rise ÷ run = 2/3.
Slope is rise over run between two points, and the y-intercept is where the line crosses the y-axis.

Often tested

  • Quadrant I (+, +), II (−, +), III (−, −), IV (+, −).
  • Slope is rise over run, never run over rise.
  • In a proportional relationship (y = kx), the unit rate k is the slope.
  • To compare functions, find each one's rate of change and its value at x = 0.

Easy to confuse: Parallel and perpendicular lines

ItemParallelPerpendicular
SlopesEqualNegative reciprocals
Example for slope 2/32/3−3/2
Product of slopesNot used−1

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedAlgebraic Problem Solving › Graphs, Slope, and Functions★★★★★

Question 1. What is the slope of the line that passes through the points (1, 3) and (4, 12)?

  1. ① 5
  2. ② 1/3
  3. ③ 3
  4. ④ −3
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ 3

Key point: Slope = (y₂ − y₁) / (x₂ − x₁)

  1. Slope = (y₂ − y₁) / (x₂ − x₁).
  2. Change in y: 12 − 3 = 9.
  3. Change in x: 4 − 1 = 3.
  4. Slope = 9 / 3 = 3.

Check: Starting at (1, 3), go right 1 and up 3 three times: (2, 6), (3, 9), (4, 12).

Wrong choices

  • 1/3: 1/3 puts the change in x over the change in y; slope is rise over run.
  • −3: −3 has the wrong sign; y goes up as x goes up, so the slope is positive.
  • 5: 5 adds the y-values (12 + 3) instead of subtracting them.
Practice questionReviewedAlgebraic Problem Solving › Graphs, Slope, and Functions★★★★★

Question 2. What are the slope and the y-intercept of the line y = −2x + 5?

  1. ① Slope −2, y-intercept −5
  2. ② Slope −2, y-intercept 5
  3. ③ Slope 2, y-intercept 5
  4. ④ Slope 5, y-intercept −2
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ② Slope −2, y-intercept 5

Key point: In y = mx + b, m is the slope and b is the y-intercept

  1. Compare with y = mx + b.
  2. The number multiplied by x is m = −2, the slope.
  3. The number added on is b = 5, the y-intercept.

Check: When x = 0, y = −2(0) + 5 = 5, so the line crosses the y-axis at (0, 5).

Wrong choices

  • Slope 5, y-intercept −2: Slope 5, y-intercept −2 switches the two numbers.
  • Slope 2, y-intercept 5: Slope 2 drops the negative sign; the line goes down from left to right.
  • Slope −2, y-intercept −5: The y-intercept is +5, not −5; at x = 0, y = 5.
Practice questionReviewedAlgebraic Problem Solving › Graphs, Slope, and Functions★★★★★

Question 3. The grid shows four labeled points, A, B, C, and D. Which point is located at (−3, 2)?

Coordinate grid from −5 to 5 on both axes with four labeled dots: A is 2 right and 3 down from the origin, B is 3 left and 2 up, C is 3 right and 2 up, and D is 3 left and 2 down.
  1. ① Point A
  2. ② Point C
  3. ③ Point B
  4. ④ Point D
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ Point B

Key point: (x, y): move left/right for x first, then up/down for y

  1. In (−3, 2), the first number is x and the second is y.
  2. x = −3: move 3 units to the left of the origin.
  3. y = 2: move 2 units up.
  4. That spot is point B.

Check: Point B is in the upper-left section, where x is negative and y is positive, which matches (−3, 2).

Wrong choices

  • Point A: Point A is at (2, −3); it switches the x and y values.
  • Point C: Point C is at (3, 2); its x-value is positive, not −3.
  • Point D: Point D is at (−3, −2); its y-value is negative, not 2.
Practice 5 questions from this unit →

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