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Quadratic Polynomial Explorer

Polynomials · Class IX

Tune a, b, c and watch y = ax^2 + bx + c bend and shift.

1
type a value
-55
0
type a value
-1010
0
type a value
-1010
Live values
  • Discriminant0
  • Vertex x0
  • Vertex y0
xy
  • y = a x^2 + b x + c

Formulas in this lab

  • Discriminant
    b2−4acb^2 - 4ac
  • Vertex x
    −b2a-\frac{b}{2a}
  • Vertex y
    c−b24ac - \frac{b^2}{4a}
Tip: When the discriminant is negative, the parabola never touches the x-axis.

Frequently asked questions

▶What does a quadratic polynomial look like?

A quadratic polynomial is $p(x) = ax^2 + bx + c$ with a non-zero. Its graph is a parabola that opens upward when a > 0 and downward when a < 0. It can have two, one, or zero real zeros depending on the discriminant $b^2 - 4ac$.

▶How do I use this lab?

Slide a, b and c to reshape the parabola. Try a = 1, b = 0, c = -4 to see a simple U cutting the x-axis at x = 2 and x = -2. Change a's sign to flip the parabola upside down.

▶Common mistake on this topic

Students often forget that the sum of zeros is $-\frac{b}{a}$ and the product is $\frac{c}{a}$. Mind the negative sign in the sum formula. Mixing these up is a frequent slip in board-exam relationship questions.

▶Where do we see parabolas in daily life?

A cricket ball hit for a six follows a parabolic path before landing. Satellite dishes used for DTH TV are shaped like parabolas to focus signals. The lab lets you model such curves by tuning a, b and c.