학습 · 대수학 1

Quadratic Functions: Parabolas, Vertex Form, and the Quadratic Formula

Every quadratic graphs as a parabola. Master vertex form to read the vertex on sight, and the quadratic formula to find the roots when factoring fails.

10 분 TEKS 6A,6B,6C,7A,7B,7C,8A 대수학 1

Every quadratic is a parabola

A quadratic function is any equation with an x² term as its highest power. Its graph is always a parabola — a U-shape that either opens up or opens down. Once you can read the equation, you can sketch the graph, find the roots, and locate the vertex without graphing software.

The anatomy of a parabola

axis of symmetry vertex (h, k) root root y-intercept
Vertex = the lowest (or highest) point. Axis of symmetry = vertical line through the vertex. Roots = where the parabola crosses x-axis.

Two equation forms — same parabola

Standard form: y = ax2 + bx + c Vertex form: y = a(x − h)2 + k In vertex form, (h, k) is the vertex. Read it directly from the equation.
The sign of a tells you everything about direction

a > 0 → opens up (vertex is the minimum).
a < 0 → opens down (vertex is the maximum).
Bigger |a| → narrower; smaller |a| → wider.

Vertex form sign trap

y = a(x − h)² + k uses minus h. So y = (x + 2)² + 4 has h = −2 (not +2), giving vertex (−2, 4). Always read the sign opposite of what's inside the parentheses.

Read the vertex from vertex form
Vertex of y = −2(x−1)² + 5?
Direction + vertex from vertex form
y=−(x+2)²+4. Opens up or down? Vertex?
Direction from a coefficient
A parabola opens downward when a is:

Vertex from standard form

y = ax2 + bx + c x-coordinate of vertex: x = −b / (2a) y-coordinate: plug that x back into the equation The same formula gives you the axis of symmetry: x = −b/(2a).

When factoring fails: the quadratic formula

Some quadratics don't factor nicely. The quadratic formula always works.

For ax2 + bx + c = 0: x = ( −b ± √(b2 − 4ac) ) / (2a) The quantity b² − 4ac is the discriminant. It tells you how many real roots there are.
Discriminant > 0
Two real roots (parabola crosses x-axis twice).
Discriminant = 0
One real root (parabola just touches x-axis at vertex).
Discriminant < 0
No real roots (parabola never crosses x-axis).

Worked example

2x2 − 5x − 3 = 0   (a=2, b=−5, c=−3) x = (5 ± √(25 + 24)) / 4 x = (5 ± √49) / 4 = (5 ± 7) / 4 x = 3   or   x = −½
Use the quadratic formula
Use quadratic formula: 2x²−5x−3=0

3-second recap

  • Standard form: y = ax² + bx + c. Vertex form: y = a(x − h)² + k.
  • Sign of a: positive opens up, negative opens down.
  • Vertex from standard: x = −b/(2a).
  • Quadratic formula: x = (−b ± √(b² − 4ac))/(2a).
  • Discriminant: b² − 4ac → tells you the number of real roots.

Check yourself

Quick check #1
What is the vertex of the parabola y = (x − 3)² + 4?
vertex(0,0)(2, −3)
Quick check #2
Solve x² − 5x + 6 = 0 by factoring.