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Quadratic Functions infographic - Roots, Vertex, and Axis of Symmetry

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Math

Quadratic Functions

Roots, Vertex, and Axis of Symmetry

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A quadratic function has the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, and its graph is always a parabola. When a>0a > 0 the parabola opens upward (minimum at the vertex); when a<0a < 0 it opens downward (maximum). The vertex represents the turning point of the parabola, and the axis of symmetry - a vertical line through the vertex - divides the parabola into two mirror-image halves.

Quadratics appear throughout physics (projectile motion), economics (profit maximization), and engineering (cable shapes in suspension bridges). The quadratic formula, the discriminant, and factoring methods all give different views of the same object: where the parabola crosses the x-axis and what shape it takes.

Key Facts

  • Standard form: f(x)=ax2+bx+cf(x) = ax^2 + bx + c
  • Vertex form: f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, vertex at (h,k)(h, k)
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
  • Axis of symmetry: x = -b/(2a)
  • Discriminant Δ=b2−4ac\Delta = b^2 - 4ac: Δ>0\Delta > 0 two real roots; Δ=0\Delta = 0 one root; Δ<0\Delta < 0 no real roots
  • Sum of roots = −ba-\frac{b}{a}; product of roots = ca\frac{c}{a}.

Vocabulary

Vertex
The maximum or minimum point of a parabola, located at (−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right).
Axis of symmetry
The vertical line x = -b/(2a) that divides the parabola into two symmetric halves.
Discriminant
The expression b² - 4ac under the square root in the quadratic formula; determines the number and type of roots.
Root (zero)
An x-value where f(x)=0f(x) = 0; the x-intercept(s) of the parabola.
Vertex form
f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, which directly shows the vertex (h,k)(h, k) and direction of opening.

Common Mistakes to Avoid

  • Forgetting that the axis of symmetry is x=−b2ax = \frac{-b}{2a}, not x=b2ax = \frac{b}{2a}. The negative sign is easy to drop.
  • Thinking a negative discriminant means 'no answer.' The equation has two complex (imaginary) roots - just no real ones.
  • Expanding vertex form incorrectly. (x - h)² ≠ x² - h². The correct expansion is x² - 2hx + h².
  • Misidentifying the vertex as the y-intercept. The y-intercept is the constant c in standard form; the vertex is at a different x-value.

Practice Questions

  1. 1 Find the vertex and roots of f(x)=x2−4x−5f(x) = x^2 - 4x - 5. Does the parabola open up or down?
  2. 2 A ball's height is h(t) = -5t² + 20t + 1 meters. When does it reach maximum height and how high is that?
  3. 3 Convert f(x)=2x2−8x+6f(x) = 2x^2 - 8x + 6 to vertex form and identify the vertex.