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➕ Algebra 1 · Linear Functions

Graphing Linear Equations: notes and common mistakes

Study notes

Key ideas

  • The graph of an equation in two variables is the set of all its solutions plotted as points. For a linear equation it is a straight line.
  • A point is on the line exactly when its coordinates make the equation true.
  • Exactly one straight line passes through two different points, so two points draw the line and a third checks your work.
  • Three ways to find points: a table of values, the y-intercept plus the slope, and the two intercepts.
  • y = b is a horizontal line (slope 0). x = a is a vertical line (slope undefined).

Steps

  1. Choose a method that fits the form of the equation.
  2. Find at least two points, and a third to check.
  3. Plot the points (x first, then y) and see whether they line up.
  4. Draw a straight line through them with arrowheads on both ends.

Formulas

  • Slope-intercept form: y = mx + b: slope m, y-intercept b (the point (0, b))
  • y-intercept: Set x = 0 and solve for y
  • x-intercept: Set y = 0 and solve for x
  • Horizontal line: y = b, slope 0
  • Vertical line: x = a, slope undefined

Watch out

  • Plot x first, then y: (3, −2) is right 3 and down 2, not up 3 and left 2.
  • A negative slope goes down as you move right (or up as you move left).
  • When you substitute a negative number, use parentheses: −2(−3) + 5.
  • A line through the origin has both intercepts equal to 0, so it crosses both axes at (0, 0) and you need another point.
  • y = 3 is horizontal, not vertical. Think HOY VUX.

Why it works

  • Why is the graph of y = mx + b a line? Each time x goes up by 1, y changes by the same amount m. A constant rise over run means a constant steepness, which is what makes a line straight.
  • Why do two points draw the line? Exactly one straight line passes through two different points. The third point only checks your arithmetic.
  • Why set x = 0 for the y-intercept? Every point on the y-axis has x-coordinate 0.

Common mistakes

✗ Mistakey = 5 is a vertical line.
✓ Insteady = 5 is horizontal, because every point has y-coordinate 5. The vertical line is x = 5. Think HOY VUX.
✗ MistakeA slope of −4/3 means go up 4 and right 3.
✓ InsteadThe negative sign means down as you go right: down 4 and right 3 (or up 4 and left 3).
✗ MistakeTo find the x-intercept, set x = 0.
✓ InsteadThe x-intercept is the x-value where the line crosses the x-axis, where y = 0. Set y = 0 and solve for x. Setting x = 0 finds the y-intercept.
✗ MistakeTo plot (2, 5), go up 2 and right 5.
✓ InsteadThe x-coordinate comes first: go right 2, then up 5.

Teach it back

Explain to a friend how to graph x + 2y = 8 in two ways (with the intercepts, and with slope-intercept form), and how to check that (2, 3) is on the line.

Explaining it out loud shows what you really know. Did you…

  • I found the x-intercept 8 and the y-intercept 4 by setting each variable to 0 in turn, which gives the points (8, 0) and (0, 4).
  • I solved for y to get y = −(1/2)x + 4 and used the slope −1/2 (down 1, right 2) from (0, 4).
  • I checked (2, 3) by substituting: 2 + 2(3) = 8, so it is on the line.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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