9.3 Quadratic Functions: Key Features and Graphs

9.3 Quadratic Functions: Key Features and Graphs

Quick Answer: A quadratic function is any function of the form f(x) = ax^2 + bx + c with a != 0; its graph is a parabola. On the Florida Algebra 1 EOC (B.E.S.T.), MA.912.AR.3.7 expects you to identify and interpret the vertex, axis of symmetry, zeros, maximum or minimum, y-intercept, domain, and range; MA.912.AR.3.8 expects you to graph quadratics from standard, vertex, and factored forms and identify transformations from vertex form.

Three Useful Forms

• Standard form f(x) = ax^2 + bx + c: y-intercept is (0, c); direction by sign of a.
• Vertex form f(x) = a(x - h)^2 + k: vertex (h, k); axis of symmetry x = h; direction and stretch by a.
• Factored form f(x) = a(x - r1)(x - r2): zeros (r1, 0) and (r2, 0); y-intercept is f(0) = a * r1 * r2.

Moving between forms is a recurring EOC skill: standard to vertex by completing the square; standard to factored by factoring; vertex to standard by expanding.

Key Features (MA.912.AR.3.7)

Vertex

The vertex is the highest or lowest point on the parabola. From standard form, its x-coordinate is x = -b/(2a); substitute back into f(x) to find the y-coordinate. From vertex form, the vertex is (h, k) - read it directly, but watch the sign: f(x) = a(x - 3)^2 + 5 has vertex (3, 5), not (-3, 5).

Worked example. Find the vertex of f(x) = x^2 - 8x + 15. x = -(-8)/(2 * 1) = 4, and f(4) = 16 - 32 + 15 = -1, so the vertex is (4, -1).

Axis of Symmetry

The axis of symmetry is the vertical line through the vertex: x = -b/(2a) from standard form, x = h from vertex form. The parabola is a mirror image across this line.

Zeros (x-intercepts)

Zeros are the x-values where f(x) = 0. Find them by factoring, the square-root property (if b = 0), or the quadratic formula. A parabola can have two zeros, one zero (the vertex touches the x-axis), or no real zeros. The factored form shows the zeros directly as r1 and r2.

Maximum or Minimum

If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum. The vertex y-coordinate is the maximum or minimum value of the function.

Worked example. Find the maximum value of f(x) = -2x^2 + 12x - 7. Because a = -2 < 0, the vertex is a maximum. x = -12/(2 * -2) = 3, and f(3) = -18 + 36 - 7 = 11, so the maximum value is 11.

y-Intercept

The y-intercept is f(0). From standard form it is c. From factored form it is a * r1 * r2; from vertex form it is a * h^2 + k. EOC items often ask for the y-intercept when the equation is in vertex or factored form to test whether you can evaluate at x = 0 instead of converting first.

Domain and Range

Domain of any unrestricted quadratic: all real numbers. Range: if a > 0, [k, infinity) where k is the vertex y-coordinate (minimum); if a < 0, (-infinity, k] where k is the vertex y-coordinate (maximum). In a real-world context, the domain is often restricted (time >= 0, length > 0), and the range is the corresponding output values - not the full theoretical range.

EOC Trap Summary

• Sign of h. Read h as the opposite of what is inside the parentheses: (x - 3) means h = 3, shift right.
• Reading the vertex with the wrong sign. f(x) = a(x - 5)^2 - 2 has vertex (5, -2), not (-5, -2).
• Confusing zeros with the y-intercept. Zeros are x-values where y = 0; the y-intercept is where x = 0.
• Reporting the full theoretical range in a context. Projectile and area problems restrict domain and range.

Graphing from Each Form (MA.912.AR.3.8)

A complete EOC graph shows: vertex, axis of symmetry (dashed vertical line), both zeros (if real), y-intercept, and one or two additional symmetric points.

• From standard form: x = -b/(2a) for the axis and vertex; set f(x) = 0 for zeros; read c for the y-intercept.
• From vertex form: Plot the vertex (h, k); the axis is x = h; find one more point by evaluating f at a convenient x and reflect it; the y-intercept is f(0).
• From factored form: Plot the zeros (r1, 0) and (r2, 0); the axis is x = (r1 + r2)/2, and the vertex lies on that line; the y-intercept is a * r1 * r2.

Transformations of Quadratic Functions

From the parent f(x) = x^2, vertex form f(x) = a(x - h)^2 + k describes transformations:

• a > 1 or a < -1: vertical stretch (narrower).
• 0 < a < 1 or -1 < a < 0: vertical compression (wider).
• a < 0: reflection across the x-axis (opens down).
• h > 0: shift right by h units. h < 0: shift left by |h| units.
• k > 0: shift up by k units. k < 0: shift down by |k| units.

The sign trap: in f(x) = (x - 3)^2, h = 3, so the graph shifts right 3, not left. The subtraction inside the parentheses produces a right shift. Likewise f(x) = (x + 3)^2 is f(x) = (x - (-3))^2, so h = -3 and the graph shifts left 3. EOC items include the "left instead of right" distractor.