Answer Key: Quadratic Functions — Graphs, Zeros, and the Vertex

f(0) = (0)² − 4(0) + 3 = 0 − 0 + 3 = 3.   f(2) = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. Full credit for squaring first, then multiplying, then combining. Watch-for: a student who computes (−4·2) before squaring correctly, or who writes (2)² = 4 but then does 4 − 4 = 0 and forgets the −8; anchor the order: square, multiply, add/subtract left to right.

Two numbers that multiply to 6 and add to 5 are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3). Setting each factor to 0: x + 2 = 0 → x = −2; x + 3 = 0 → x = −3. Watch-for: sign confusion — because both middle and last terms are positive, both factors are (x + …); the roots are negative.

f(4) = (4)² − 4(4) + 3 = 16 − 16 + 3 = 3. It equals f(0) = 3 because x = 0 and x = 4 are the same distance (2 units) on each side of the axis of symmetry x = 2; symmetric inputs have equal outputs. Alternate: read (4, 3) as the mirror of (0, 3) straight from the graph or table.

4a. Zeros are where y = 0: x = −1 and x = 3 (the points (−1, 0) and (3, 0)).
4b. Lowest point in the table is (1, −4), so the vertex is (1, −4). Because the parabola opens up, it is a minimum.
4c. Axis of symmetry: x = 1. The two zeros −1 and 3 are equally spaced around it — the midpoint (−1 + 3) ÷ 2 = 1 is the axis. (Also visible because (0, −3) and (2, −3) share a y-value and straddle x = 1.)
4d. y-intercept: y = −3 (the value at x = 0, the point (0, −3)).
Background (not required from students): this parabola is y = x² − 2x − 3 = (x + 1)(x − 3); vertex x = −(−2) ÷ 2 = 1, f(1) = 1 − 2 − 3 = −4. ✔

Two numbers that multiply to 10 and add to −7 are −2 and −5: x² − 7x + 10 = (x − 2)(x − 5) = 0. Zero Product Property: x − 2 = 0 → x = 2; x − 5 = 0 → x = 5. Axis of symmetry is halfway between the roots: x = (2 + 5) ÷ 2 = 3.5. Check: −b ÷ (2a) = 7 ÷ 2 = 3.5. ✔

6a. Since a = −16 < 0, the parabola opens down, so the vertex is a maximum (a highest point).
6b. Time of vertex: t = −b ÷ (2a) = −32 ÷ (2·−16) = −32 ÷ −32 = 1 second. Greatest height: h(1) = −16(1)² + 32(1) = −16 + 32 = 16 feet. Interpretation: the ball reaches its highest point of 16 feet at 1 second after the kick. Alternate: the zeros are t = 0 and t = 2 (factor −16t(t − 2)); their midpoint t = 1 is the vertex time. Watch-for: dropping the negative on a and getting the wrong sign, or stopping after finding t = 1 without evaluating the height.

The mistake: The student factored as (x − 2)(x − 3), but that multiplies out to x² − 5x + 6, not x² − x − 6. They ignored the negative constant −6: the two numbers must multiply to −6 (so they have opposite signs) and add to −1.
Correct work: −6 = (−3)(+2) and −3 + 2 = −1, so x² − x − 6 = (x − 3)(x + 2) = 0. Then x − 3 = 0 → x = 3; x + 2 = 0 → x = −2. The correct roots are x = 3 and x = −2.
Full credit requires (1) naming that the factors must multiply to −6 (opposite signs), not +6, and (2) the correct roots 3 and −2. Sense-check to offer: substitute x = 2 into the original: 4 − 2 − 6 = −4 ≠ 0, so 2 is not a root; x = 3: 9 − 3 − 6 = 0 ✔.

8a. For y = 2(x − 3)² − 5, the form y = a(x − h)² + k gives h = 3 and k = −5, so the vertex is (3, −5) and the axis of symmetry is x = 3. Since a = 2 > 0, it opens up and the vertex is a minimum. Watch-for: the sign of h — the form uses (x − h), so (x − 3) means h = +3, not −3.
8b. A(x) = −2x² + 40x, so a = −2, b = 40. Vertex width: x = −b ÷ (2a) = −40 ÷ (2·−2) = −40 ÷ −4 = 10 feet. Maximum area: A(10) = −2(10)² + 40(10) = −200 + 400 = 200 square feet. Interpretation: a width of 10 ft (with length 40 − 2(10) = 20 ft) gives the largest pen, 200 sq ft. Because a < 0, the vertex is a maximum. Alternate: zeros of A are x = 0 and x = 20 (factor −2x(x − 20)); midpoint x = 10 is the vertex.

Model answer. Claim: The classmate is wrong. The roots are 2 and 4, but the vertex is at x = 3, not x = 4. Evidence: Factor: x² − 6x + 8 = (x − 2)(x − 4) = 0, so the roots are x = 2 and x = 4. Axis/vertex x = −b ÷ (2a) = −(−6) ÷ (2·1) = 6 ÷ 2 = 3; the vertex is (3, f(3)) = (3, 9 − 18 + 8) = (3, −1). Reasoning: The axis of symmetry sits halfway between the two roots: (2 + 4) ÷ 2 = 3. A root is a point where y = 0 (the curve crosses the x-axis), but the vertex is where the curve turns — it is the midpoint of the two roots, not one of the roots themselves. The classmate mixed up a root (x = 4) with the axis (x = 3).
Note: Full credit requires the correct roots (2 and 4), the correct vertex x (3), and a clear statement that the axis is the midpoint of the roots — so the classmate is wrong.

Articulate: Accept any correct explanation: "A zero (root) is an x-value where y = 0 — where the parabola crosses the x-axis. The vertex is the single point where the curve turns (its lowest or highest point). A parabola can have two zeros but only one vertex; the zeros cross the axis, the vertex sits halfway between them."
Connect: Valid items where a vertex was used as a max/min in context include Q6 (greatest height, a maximum) and, on block, Q8b (maximum area). Either earns credit if correctly named.
Extend: Any genuine max/min situation — e.g., the maximum revenue as ticket price changes, the minimum cost of production, the peak height of a fireworks shell, or the largest rectangular area for a fixed perimeter. Must clearly identify the quantity being maximized or minimized at the vertex.

Sample (roots 2 and 5): factored form (x − 2)(x − 5). (2) Multiply out: x² − 7x + 10, so y = x² − 7x + 10. (3) Axis: x = (2 + 5) ÷ 2 = 3.5 (or −b ÷ (2a) = 7 ÷ 2 = 3.5); vertex: f(3.5) = 12.25 − 24.5 + 10 = −2.25, so (3.5, −2.25). (4) A reasonable table: (2, 0), (3, −2), (3.5, −2.25), (4, −2), (5, 0), symmetric around x = 3.5. Full credit requires a correct factored form, a correct standard form, an axis halfway between the two chosen roots, a matching vertex, and a symmetric table/sketch. Many correct answers exist depending on the roots chosen.