Quadratic Functions in Depth

You have seen parabolas before. You know that $y = x^2$ creates a U-shaped curve, and you have probably solved a few quadratic equations. But now it is time to truly master these curves - to understand them so thoroughly that you can write their equations in any form, convert between forms at will, and extract every piece of information they contain.

Think about catching a football. The ball traces an arc through the air, and if you know the right equation, you can predict exactly where it will land, how high it will go, and when it will reach its peak. That equation is a quadratic function. Architects use the same mathematics to design arches. Engineers use it to shape satellite dishes. Businesses use it to maximize profit. The parabola is everywhere, and understanding its forms gives you power over all these situations.

The key insight is this: the same parabola can be written in different ways, and each way makes different information obvious. Learning to see the same curve from multiple perspectives is what this lesson is about.

Core Concepts

Three Forms of Quadratic Functions

Every quadratic function can be written in three equivalent forms. Each form highlights different features of the parabola:

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

This is the form you encounter most often. It makes the y-intercept immediately visible (it is just $c$), and the coefficients $a$, $b$, and $c$ plug directly into the quadratic formula.

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

This form tells you the vertex at a glance - it is the point $(h, k)$. When you need to know the highest or lowest point of a parabola, vertex form is your friend.

Factored Form: $f(x) = a(x - r_1)(x - r_2)$

This form reveals the x-intercepts (also called roots or zeros) directly. The parabola crosses the x-axis at $x = r_1$ and $x = r_2$.

The value of $a$ is the same in all three forms for the same parabola. It controls the direction (up if positive, down if negative) and the width of the curve.

Vertex Form: A Closer Look

The vertex form $f(x) = a(x - h)^2 + k$ deserves special attention. The vertex is at $(h, k)$, but notice the subtraction sign in the formula. This means:

• If your function is $f(x) = (x - 3)^2 + 5$, the vertex is at $(3, 5)$
• If your function is $f(x) = (x + 2)^2 - 1$, rewrite as $(x - (-2))^2 + (-1)$, so the vertex is at $(-2, -1)$

That minus sign catches many students. When you see $(x + 2)$, that is really $(x - (-2))$, so $h = -2$.

The Axis of Symmetry

Every parabola is perfectly symmetric. If you fold it along a vertical line through the vertex, both halves match exactly. This line is the axis of symmetry.

In vertex form, the axis of symmetry is simply $x = h$.

In standard form, the axis of symmetry is $x = -\frac{b}{2a}$.

The axis of symmetry passes through the vertex. Any point on the parabola has a mirror image on the other side of this line.

Converting to Vertex Form: Completing the Square

This is one of the most important algebraic techniques you will learn. Completing the square transforms standard form into vertex form, revealing the vertex directly.

The idea is to create a perfect square trinomial - something that factors as $(x + n)^2$.

Here is the process for $f(x) = ax^2 + bx + c$:

Step 1: If $a \neq 1$, factor $a$ out of the first two terms: $$f(x) = a\left(x^2 + \frac{b}{a}x\right) + c$$

Step 2: Take half of the coefficient of $x$ inside the parentheses, and square it. This is the number you need to “complete the square.”

Step 3: Add this number inside the parentheses. But to keep the equation balanced, you must also subtract the equivalent amount (accounting for the factor of $a$ outside).

Step 4: Rewrite the perfect square trinomial as a squared binomial.

This sounds abstract, but the examples below will make it concrete.

Finding Key Features

Once you have a quadratic in any form, you can extract all its important features:

Vertex:

• From vertex form: read off $(h, k)$
• From standard form: calculate $x = -\frac{b}{2a}$, then substitute to find $y$

Axis of symmetry:

• From vertex form: $x = h$
• From standard form: $x = -\frac{b}{2a}$

Y-intercept:

• Set $x = 0$ and calculate
• In standard form, it is simply $c$

X-intercepts (if they exist):

• Set $f(x) = 0$ and solve
• In factored form, read off $r_1$ and $r_2$ directly

Maximum or minimum value:

• If $a > 0$, the parabola opens upward, so the vertex is the minimum
• If $a < 0$, the parabola opens downward, so the vertex is the maximum
• The max/min value is the y-coordinate of the vertex (that is, $k$)

Maximum and Minimum in Context

Real-world problems often ask for the maximum or minimum value of something - the highest a ball goes, the maximum profit, the minimum cost. Quadratic functions model many of these situations.

The strategy is always the same:

1. Write the quadratic function that models the situation
2. Determine whether you need a maximum (when $a < 0$) or minimum (when $a > 0$)
3. Find the vertex - the x-coordinate tells you the input that produces the max/min, and the y-coordinate is the max/min value itself

Quadratic Regression

Sometimes you have data points that follow a curved pattern, and you want to find the quadratic function that best fits the data. This is called quadratic regression.

Most graphing calculators and software can perform quadratic regression automatically. You input the data points, and the technology outputs the values of $a$, $b$, and $c$ for the best-fitting parabola.

When is quadratic regression appropriate? When your data:

• Increases then decreases (or vice versa)
• Shows a curved, roughly parabolic pattern
• Represents a phenomenon where physics or economics suggests a quadratic relationship

Review: Solving Quadratic Equations

You have multiple tools for solving $ax^2 + bx + c = 0$:

Factoring: If you can factor the expression, use the Zero Product Property. Fast when it works, but not all quadratics factor nicely.

Square Roots: If the equation has the form $(something)^2 = k$, take the square root of both sides. Remember $\pm$.

Completing the Square: Always works. Transform to vertex form, then solve.

Quadratic Formula: Always works. $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant $b^2 - 4ac$ tells you what to expect:

• Positive: two real solutions
• Zero: one real solution (a repeated root)
• Negative: no real solutions

Notation and Terminology

Term	Meaning	Example
Standard form	$f(x) = ax^2 + bx + c$	$f(x) = 2x^2 + 8x + 5$
Vertex form	$f(x) = a(x-h)^2 + k$	$f(x) = 2(x+2)^2 - 3$
Factored form	$f(x) = a(x - r_1)(x - r_2)$	$f(x) = 2(x-1)(x-5)$
Vertex	The point $(h, k)$ in vertex form; the maximum or minimum point	$(-2, -3)$
Axis of symmetry	$x = h$ or $x = -\frac{b}{2a}$; the vertical line through the vertex	$x = -2$
Completing the square	Method to convert standard form to vertex form
Roots/zeros	The x-values where $f(x) = 0$; the x-intercepts	$x = 1$ and $x = 5$
Discriminant	$b^2 - 4ac$; determines the number of real solutions

Examples

Key Properties and Rules

Form Conversion Summary

From	To	Method
Standard	Vertex	Complete the square
Standard	Factored	Factor (if possible) or find roots with quadratic formula
Vertex	Standard	Expand $(x-h)^2$ and distribute
Vertex	Factored	Set equal to zero, solve for roots
Factored	Standard	FOIL/expand and distribute $a$
Factored	Vertex	Find axis of symmetry (midpoint of roots), substitute to find vertex

The Axis of Symmetry Shortcut

When you have factored form $f(x) = a(x - r_1)(x - r_2)$, the axis of symmetry is: $$x = \frac{r_1 + r_2}{2}$$

This is the midpoint of the two x-intercepts. The vertex lies on this line.

Completing the Square - Quick Reference

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

1. Factor out $a$ from the first two terms
2. Take half of the coefficient of $x$ (inside parentheses), square it
3. Add this inside the parentheses, subtract $a$ times this value outside
4. Write the perfect square trinomial as $(x + \text{something})^2$

The vertex is $\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$.

Common Mistakes to Avoid

• Sign error in vertex form: Remember $(x - h)^2$, so $(x + 2)^2$ means $h = -2$
• Forgetting to multiply when completing the square: When $a \neq 1$, the number added inside is multiplied by $a$ when you subtract outside
• Not checking your work: Substituting a point into all forms should give the same value
• Confusing maximum and minimum: $a > 0$ means minimum (opens up); $a < 0$ means maximum (opens down)

Real-World Applications

Projectile Motion

The height of any object thrown, launched, or dropped follows a quadratic path (ignoring air resistance):

$$h(t) = -\frac{1}{2}gt^2 + v_0 t + h_0$$

where $g$ is gravitational acceleration, $v_0$ is initial velocity, and $h_0$ is initial height. The vertex tells you the maximum height and when it occurs. Setting $h(t) = 0$ tells you when the object lands.

Basketball players, quarterbacks, and golfers all intuitively understand parabolic motion - their brains calculate these arcs in real time.

Maximizing Revenue and Profit

In economics, revenue often follows a quadratic pattern. If you charge too little, you make less per item. If you charge too much, fewer people buy. Somewhere in the middle is the sweet spot.

A typical revenue function looks like $R(p) = -ap^2 + bp$, where $p$ is price. The vertex gives the price that maximizes revenue.

Minimizing Cost

Manufacturing and shipping costs often have quadratic components. Producing too few items means high per-unit costs. Producing too many means excess inventory and storage costs. The minimum-cost production level is at the vertex.

Architecture and Engineering

Parabolic arches distribute weight efficiently and appear in bridges, doorways, and cathedrals. The St. Louis Gateway Arch, suspension bridge cables (which form catenary curves, close to parabolas), and the cables of the Golden Gate Bridge all involve parabolic mathematics.

Satellite Dishes and Headlight Reflectors

A parabolic reflector has a remarkable property: signals coming from straight ahead all bounce to a single point (the focus). This is why satellite dishes and car headlights use parabolic shapes - they concentrate or project signals with perfect focus.

The equation of such a parabola is $y = \frac{1}{4p}x^2$, where $p$ is the distance from the vertex to the focus.

Self-Test Problems

Summary

• A quadratic function can be written in three forms, each revealing different information:

  • Standard form $f(x) = ax^2 + bx + c$ shows the y-intercept ($c$) and is ready for the quadratic formula
  • Vertex form $f(x) = a(x-h)^2 + k$ shows the vertex $(h, k)$ directly
  • Factored form $f(x) = a(x - r_1)(x - r_2)$ shows the x-intercepts $r_1$ and $r_2$

• Completing the square converts standard form to vertex form. Factor out $a$, take half of the linear coefficient and square it, add inside and subtract the equivalent outside, then rewrite as a squared binomial.

• The vertex is the maximum or minimum point of the parabola. In standard form, find it with $x = -\frac{b}{2a}$, then substitute to get the y-coordinate.

• The axis of symmetry is $x = h$ (vertex form) or $x = -\frac{b}{2a}$ (standard form). It passes through the vertex.

• For optimization problems, set up a quadratic function, find the vertex, and interpret: the x-coordinate is the input that produces the maximum or minimum, and the y-coordinate is that max/min value.

• The value of $a$ determines direction: $a > 0$ means the parabola opens upward (minimum at vertex); $a < 0$ means it opens downward (maximum at vertex).

• Quadratic functions model projectile motion, revenue optimization, cost minimization, and the geometry of parabolic reflectors.

You now have complete mastery over the parabola. You can see it from any angle - standard, vertex, or factored - and move between perspectives at will. Whether you are predicting where a ball will land, maximizing profit, or designing a satellite dish, the parabola is your tool. The techniques you have learned here - especially completing the square and optimization - will serve you throughout calculus and beyond.