Roots, Intercepts, Symmetry, and Graphs

6.5: Roots, Intercepts, Symmetry, and Graphs. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 83 parts Practice Flashcards

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Roots and intercepts

Objective: Read zeros from a factored quadratic.

Prerequisite skills: roots, vertex form, and graph interpretation. An x-intercept occurs where y = 0. In factored form, set each variable factor to zero. The y-intercept instead comes from x = 0; do not confuse the two kinds of intercept. Common error: using the factor constants with their displayed signs as roots. Restrictions and interpretation: Real roots correspond to x-intercepts; the axis of symmetry is vertical.

Formulas

$a(x − r)(x − s) = 0 ⇒ x = r or s$

Worked examples

Try itWhat is the greater x-intercept of y = (x − 23)(x − 27)?Show answer

Setting y=0 gives x=23 or x=27. The larger x-intercept is 27; its coordinate pair is (27,0). Therefore the requested answer is 27.

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Roots and intercepts

Objective: Read zeros from a factored quadratic.

Formulas

$a(x − r)(x − s) = 0 ⇒ x = r or s$

Worked examples

Try itWhat is the greater x-intercept of y = (x − 23)(x − 27)?Show answer

Setting y=0 gives x=23 or x=27. The larger x-intercept is 27; its coordinate pair is (27,0). Therefore the requested answer is 27.

Axis of symmetry

Objective: Average the roots to find the axis of a parabola.

The vertex lies halfway between two real roots because the parabola is symmetric. Equivalently, its horizontal coordinate is −b/(2a) in standard form. The axis is a vertical line x = h. Common error: using half the distance between roots as the vertex coordinate. Restrictions and interpretation: Real roots correspond to x-intercepts; the axis of symmetry is vertical.

Formulas

$h = (r + s)/2 = −b/(2a)$

Worked examples

Try itThe roots of a quadratic are 23 and 29. What is the x-coordinate of its axis of symmetry?Show answer

Average the roots: h = (23+29)/2 = 26. The axis is x = 26, not merely half the distance 3. Therefore the requested answer is 26.

Recover a graph parameter

Objective: Use one point to determine a quadratic multiplier.

The roots determine a factored expression only up to a nonzero multiplier a. Substitute another known point to solve for that multiplier. The point must not be a root, or it provides no information about a. Common error: assuming a quadratic with given roots must have leading coefficient 1. Restrictions and interpretation: Real roots correspond to x-intercepts; the axis of symmetry is vertical. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$y₀ = a(x₀ − r)(x₀ − s)$

Worked examples

Try itA quadratic with roots 24 and 26 passes through (0, 15600). In y = a(x − 24)(x − 26), what is a?Show answer

At x=0, the product is (−24)(−26) = 624. Thus 15600 = a(624), giving a = 25. It is nonzero, so the function is quadratic. Therefore the requested answer is 25.

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