Introduction to Quadratic Functions

Welcome to the world of Quadratic Functions! While linear functions (straight lines) are great for simple relationships, quadratics allow us to model the curved paths of the world around us. Whether you are watching a basketball arc toward a hoop, a satellite dish reflecting signals, or a company trying to find its "sweet spot" for maximum profit, you are looking at a quadratic function in action.

In this chapter, we will learn how to recognize different "outfits" (forms) quadratic functions wear, how to find their key features, and how to solve equations using a powerful tool called the discriminant. Don't worry if it seems like a lot of algebra at first—we'll break it down step-by-step!

1. The Three Faces of a Quadratic

A quadratic function is a function where the highest power of \(x\) is \(2\). Depending on what information you have (or what you need to find), you can write it in three different ways. Think of these as different "modes" for the same math.

Form 1: Standard Form

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

  • Why use it? It's the most common way to see a quadratic.
  • The Secret: The value of \(c\) is always your \(y\)-intercept. If you plug in \(x = 0\), everything disappears except \(c\)!
  • The Shape: If \(a > 0\), the graph is a "smiley face" (opens upward). If \(a < 0\), it's a "frowny face" (opens downward).

Form 2: Factorized (Intercept) Form

\(f(x) = a(x - p)(x - q)\)

  • Why use it? It tells you exactly where the graph crosses the \(x\)-axis.
  • The Secret: The values \(p\) and \(q\) are your \(x\)-intercepts (also called roots or zeros). Note: Be careful with the signs! If the bracket is \((x - 3)\), the intercept is \(+3\).

Form 3: Vertex Form

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

  • Why use it? It gives you the "turning point" of the graph immediately.
  • The Secret: The point \((h, k)\) is the vertex. Just like before, the sign of \(h\) inside the bracket is flipped.

Quick Takeaway: Switching between these forms is just a matter of expanding brackets or completing the square. Each form has a special "superpower" for finding a specific feature of the graph!

2. Finding Key Features

To sketch a quadratic graph (a parabola), you need a few "landmarks." Here is how to find them using the Standard Form \(ax^2 + bx + c\):

  • \(y\)-intercept: Set \(x = 0\). Result: \((0, c)\).
  • Axis of Symmetry: This is the invisible mirror line through the middle. Use the formula: \(x = -\frac{b}{2a}\).
  • The Vertex: This is the highest or lowest point. The \(x\)-coordinate is \(-\frac{b}{2a}\). To find the \(y\)-coordinate, just plug that \(x\) value back into your function.
  • \(x\)-intercepts: Set \(f(x) = 0\) and solve for \(x\). (More on this in the next section!)

Did you know? The axis of symmetry is always exactly halfway between the two \(x\)-intercepts. If you know the intercepts are \(2\) and \(8\), the mirror line must be at \(x = 5\)!

3. Solving Quadratic Equations

Solving \(ax^2 + bx + c = 0\) means finding the "roots" or "zeros"—the places where the graph touches the floor (\(x\)-axis).

Method A: Factoring

If you can find two numbers that multiply to give \(ac\) and add to give \(b\), you can factor the equation. This is usually the fastest method for Paper 1 (no calculator).

Method B: The Quadratic Formula

If factoring looks too hard, use the "Old Faithful" formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Common Mistake: Forgetting that the \(2a\) goes under the entire numerator, not just the square root!

Method C: Technology (GDC)

On Paper 2, don't waste time! Use your "Polynomial Solver" or graph the function and find the "zeros" using the calculation menu. This is a vital skill for IB students.

4. The Discriminant: The "Crystal Ball"

Before you even solve an equation, you can predict what the roots will look like using the discriminant. The symbol for this is the Greek letter Delta: \(\Delta\).

\(\Delta = b^2 - 4ac\)

Depending on the value of \(\Delta\), you get three different "natures of roots":

  1. If \(\Delta > 0\): There are two distinct real roots. The graph crosses the \(x\)-axis twice.
  2. If \(\Delta = 0\): There is one repeated real root. The vertex of the graph sits exactly on the \(x\)-axis (it "kisses" the axis).
  3. If \(\Delta < 0\): There are no real roots. The graph is floating entirely above or below the \(x\)-axis.

Key Takeaway: If an IB question asks you to "find the values of \(k\) for which the equation has two real roots," it is a hidden signal to use \(\Delta > 0\).

5. Quadratic Inequalities

Sometimes you need to find where a quadratic is "greater than zero" (\(f(x) > 0\)) or "less than zero" (\(f(x) < 0\)).

Step-by-Step Process:

  • 1. Find the roots of the equation \(f(x) = 0\).
  • 2. Draw a quick sketch of the parabola.
  • 3. If you want \(> 0\), look for the parts of the curve above the \(x\)-axis.
  • 4. If you want \(< 0\), look for the parts below the \(x\)-axis.

Example: For a smiley face graph with roots at \(1\) and \(5\), the part between them is below the axis (\(1 < x < 5\)), and the parts outside them are above the axis (\(x < 1\) or \(x > 5\)).

6. Higher Level (HL) Extensions

If you are an HL student, you will also explore AHL 2.12, which connects quadratics to the Sum and Product of Roots.

For the equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):

  • Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
  • Product of roots: \(\alpha\beta = \frac{c}{a}\)

This is extremely useful when you are asked to create a new quadratic equation based on the roots of an old one without actually solving them!

Summary Checklist

  • Can I identify \(a, b,\) and \(c\) from an equation?
  • Do I know which way the parabola opens?
  • Can I find the vertex using \(x = -\frac{b}{2a}\)?
  • Can I calculate the discriminant and explain what it means?
  • Do I know how to use my GDC to find intersections and zeros?

Don't forget: Practice is the key to mastering quadratics. Start with basic factoring and move toward complex discriminant problems. You've got this!