Graphing Linear Equations — Slope and Intercepts

ACT Math · Axiom Academy

LESSON Graphing Linear Equations Slope, intercepts, and the equation of a line — the toolkit the ACT reuses in nearly every coordinate-geometry question. Slope measures steepness: how far a line rises (or falls) for every step it runs to the right. Given two points (x_1,y_1) and (x_2,y_2) , the slope is the change in y divided by the change in x . Example: find the slope through (1,2) and (5,8) . Slope — rise 3, run 2, for every step right. Horizontal line, no rise. m=0 . Vertical line — run =0 , division by zero. 2. Slope-Intercept Form — Graphing from the Equation The most common line format on the ACT writes y directly in terms of x : slope-intercept form . Once you can read m and b off the equation, graphing takes two moves — plot b , then step by the slope. Step 1 — plot the y-intercept: b=-1 , so start at (0,-1) . Step 2 — step by the slope : right 2, up 1, landing at (2,0) . Repeat the same step (right 2, up 1) to extend the line either direction. Set x=0 — whatever's left is b . For y=2x+5 : y=2(0)+5=5 , so the y-intercept is (0,5) . Set y=0 and solve for x . For y=2x+5 : , so the x-intercept is . 3. Point-Slope Form — When You Don't Have b Yet Sometimes a question hands you a slope and any point on the line — not the y-intercept. Point-slope form uses that point directly, without needing b first. (x_1,y_1) is any known point on the line Example: the line through (3,-1) with slope -2 . Slope-intercept: y=-2x+5 — matches (3,-1) since -2(3)+5=-1 . ✓︎

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