2 · IB Maths
Quadratic functions
Slide a, b and c and watch the vertex move and the roots appear, merge and vanish, with the discriminant predicting every change before it happens.
Predict first
Start at a = 1.0, b = −2.0, c = −1.5, which has two real roots. If you raise c to 1.0, what happens to the graph?
Δ = b² − 4ac = 4 − 4(1)(1) = 0, so there is one repeated root at the vertex and the x-axis is a tangent.
Check it in the model: Move the c slider from −1.5 to 1.0 and read the Discriminant Δ and Roots in the readout.
y = ax² + bx + c vertex roots y-intercept
Presets
y = ax² + bx + c (a ≠ 0)
Δ = b² − 4ac
x = (−b ± √Δ) ÷ 2a
vertex: x = −b ÷ 2a, y = c − b² ÷ 4a
Δ = b² − 4ac
x = (−b ± √Δ) ÷ 2a
vertex: x = −b ÷ 2a, y = c − b² ÷ 4a
Try this
- Start from the Two roots preset and raise c until the roots merge. What value of c gives Δ = 0, and what is the vertex height then?
- Set a = 1 and b = 0, then change c from −4 to +4. How do the roots move, and where is the axis of symmetry?
- Press the Frown preset and make a negative. Which sign of a gives a maximum, and how does the vertex y-value compare with c?
Exam tip: Check Δ = b² − 4ac before solving: Δ > 0 gives two real roots, Δ = 0 a repeated root with the x-axis tangent, and Δ < 0 none, so a tangent-line question is really a Δ = 0 equation.