Quadratic functions & equations
The idea
A quadratic function has the form f(x) = ax² + bx + c with a ≠ 0, and its graph is a parabola — the shape traced by thrown balls, satellite dishes, and profit curves. You already know functions, exponents, and algebraic manipulation; quadratics add a toolkit of equivalent forms. Standard form shows the y-intercept, factored form shows the roots, and vertex form a(x − h)² + k shows the maximum or minimum. Much of the skill is choosing the form that exposes what you need.
To solve ax² + bx + c = 0 you can factor, complete the square, or use the quadratic formula x = (−b ± √(b² − 4ac))/(2a), which always works. The discriminant b² − 4ac announces in advance how many real solutions exist: two when positive, one when zero, none when negative. The frequent misconception is that the vertex sits at a root or at the y-intercept; in fact it lies at x = −b/(2a), on the axis of symmetry, exactly halfway between the roots whenever roots exist.
Worked example
A ball is launched from a platform so that its height after t seconds is h(t) = −5t² + 20t + 25 meters. Find the maximum height and the time when the ball hits the ground.
- The coefficient of t² is negative, so the parabola opens downward and the vertex is the maximum. The vertex time is t = −b/(2a) = −20/(2 × (−5)) = 2 s.
- Evaluate the height there: h(2) = −5(4) + 20(2) + 25 = −20 + 40 + 25 = 45, so the peak is 45 m.
- The ball lands when h(t) = 0. Divide −5t² + 20t + 25 = 0 by −5 to get the cleaner equation t² − 4t − 5 = 0.
- Factor: t² − 4t − 5 = (t − 5)(t + 1), so t = 5 or t = −1. A negative time is before launch, so reject t = −1.
- Sanity check: the ball takes 2 s to rise from 25 m to 45 m but 3 s to fall from 45 m all the way to 0 m. The longer descent covers more height, so the asymmetry makes physical sense.
Answer. The ball reaches a maximum height of 45 m at t = 2 s and hits the ground at t = 5 s.
Check your understanding
- How does the discriminant reveal the number of real solutions before you finish solving, and what does each case look like on the graph?
- Why is the vertex halfway between the two roots when they exist, and what breaks in that statement when there are no real roots?
- When would you choose completing the square over the quadratic formula, given that both always work?
- How would the landing time change if the same ball were launched from the ground instead of from a 25 m platform?
Build the foundations first
Quadratic functions & equations builds on these concepts. If any feel shaky, start there.