Controls the curve
A positive value opens the parabola upward; a negative value flips it downward. Larger values of |a| make the curve narrower.
An interactive parabola explorer
Explore the shape of a quadratic
Move the numbers.
Watch the curve.
Change a, b, and c to see how each coefficient reshapes the graph—and what it means for the roots.
Live graph
Opens upward
roots vertex
Meet the coefficients
Three numbers,
three different jobs.
Try moving just one slider at a time. Keeping the other two fixed makes each coefficient’s effect much easier to spot.
Moves the turning point
Changing b slides the axis of symmetry, moving the vertex left or right while also changing its height.
Sets the starting height
Because y = c when x = 0, this is where the graph crosses the vertical axis. Move it to lift or lower the curve.
What the formula tells us
Read the curve
The discriminant b² − 4ac predicts how many times the parabola meets the x-axis.
Discriminant
16Two distinct real roots
Roots
-1 · 3The x-values where y equals zero
Vertex
(1, -4)The curve’s lowest point
The quadratic formula
For ax² + bx + c = 0
x =−b ± √b² − 4ac2a
Substitute your values for a, b, and c. The ± creates the two possible roots—the graph’s x-intercepts.