Quadratic Functions — Everything You Need to Know

Quadratic functions are one of the most important foundations in algebra. Once you truly understand them, a huge range of other problem types become much more approachable.

General form​

A quadratic function takes the form:

$f(x) = ax^2 + bx + c \quad (a \neq 0)$

Its graph is a parabola — opening upward when $a > 0$, downward when $a < 0$.

3 important representations​

1. Standard form​

$f(x) = ax^2 + bx + c$

Easy to identify coefficients and find the $y$-intercept at $(0, c)$.

2. Vertex form​

$f(x) = a(x - h)^2 + k$

The vertex of the parabola is at $V(h, k)$. Makes it easy to spot the maximum or minimum value.

3. Factored form​

$f(x) = a(x - x_1)(x - x_2)$

Used when the two roots $x_1, x_2$ are known. Immediately reveals the $x$-intercepts.

Formulas you must know​

$x$-coordinate of the vertex: $x_V = -\frac{b}{2a}$

$y$-coordinate of the vertex: $y_V = f(x_V) = c - \frac{b^2}{4a}$

Discriminant: $\Delta = b^2 - 4ac$

• $\Delta > 0$: two distinct real roots $x_{1,2} = \dfrac{-b \pm \sqrt{\Delta}}{2a}$
• $\Delta = 0$: one repeated root $x = -\dfrac{b}{2a}$
• $\Delta < 0$: no real roots

Quick graphing technique​

To sketch a parabola accurately, just identify 5 points:

1. Vertex $V(h, k)$
2. $y$-intercept: $(0, c)$
3. The point symmetric to $(0, c)$ across the axis of symmetry
4. $x$-intercepts (if any): $x_1, x_2$

The axis of symmetry is the vertical line $x = h = -\dfrac{b}{2a}$.

Real-world applications​

Quadratic functions describe many natural phenomena:

• Physics: projectile motion, free fall under gravity
• Engineering: suspension bridge design, parabolic antennas
• Economics: revenue optimization, break-even analysis

Example: A ball is thrown upward with initial velocity $v_0 = 20$ m/s from height $h_0 = 1.5$ m. Its height over time:

$h(t) = -5t^2 + 20t + 1.5$

The vertex gives the maximum height and the time at which it occurs.

Practice with MathPal​

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