# Quadratic functions, roots, and vertices

> IB Mathematics: Applications & Interpretation SL · Unit 2: Functions
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u2-quadratic-functions-roots-and-vertices/

Quadratic functions are second-order polynomials that model parabolic motion, profit curves, and projectile paths. This module explains how to find their roots (x-intercepts) and vertices, the two most important features for solving applied IB AI SL problems.

**Prerequisites:** [Polynomial basics](https://www.owlsprep.com/study/ib-math-ai-sl-u1-polynomial-operations/); [Solving linear equations](https://www.owlsprep.com/study/ib-math-ai-sl-u1-linear-equations/)

## Learning objectives

- Identify key features of quadratic functions across different forms
- Calculate roots of quadratic functions using multiple methods
- Find the vertex of a quadratic and identify it as maximum or minimum
- Interpret roots and vertices in real-world applied contexts

## Forms of Quadratic Functions

**Quadratic function** — A second-degree polynomial function whose graph is a parabola. It opens upwards if $a>0$ (minimum vertex) and downwards if $a<0$ (maximum vertex).

*Notation:* $f(x) = ax^2 + bx + c, a \neq 0$

*Example:* $f(x) = 2x^2 - 5x + 2$ is a quadratic opening upwards.

- **Standard form**: $f(x) = ax^2 + bx + c$, directly gives the y-intercept $(0, c)$
- **Factored form**: $f(x) = a(x - r_1)(x - r_2)$, directly gives the roots $r_1, r_2$
- **Vertex form**: $f(x) = a(x - h)^2 + k$, directly gives the vertex $(h, k)$

**Worked example:** Write $f(x) = x^2 - 4x + 3$ in factored and vertex form, and state what feature each form gives directly.

1. Factor the quadratic: find two numbers that multiply to 3 and add to -4, which are -1 and -3. This gives factored form:
2. $$f(x) = (x - 1)(x - 3)$$
3. Convert to vertex form by completing the square: group x terms and complete the square:
4. $$f(x) = (x^2 - 4x) + 3 = (x^2 - 4x + 4) - 4 + 3 = (x - 2)^2 - 1$$
5. Factored form gives roots at $x=1$ and $x=3$ directly, and vertex form gives the vertex at $(2, -1)$ directly.

> **Exam tip:** Always check what form of quadratic you are given to avoid unnecessary calculations for features you can read directly.

## Finding Roots of Quadratic Functions

**Root of a quadratic** — A value of $x$ for which $f(x) = 0$, corresponding to where the parabola crosses the x-axis. A quadratic can have 0, 1, or 2 real roots.

*Example:* The roots of $f(x) = (x-1)(x-3)$ are $x=1$ and $x=3$.

> **info**
>
> The discriminant $\Delta = b^2 - 4ac$ tells you the number of real roots: $\Delta > 0$ (2 distinct roots), $\Delta = 0$ (1 repeated root), $\Delta < 0$ (no real roots).

**Worked example:** Find all real roots of $f(x) = 2x^2 - 7x + 3$.

1. Identify coefficients from standard form: $a=2$, $b=-7$, $c=3$. Calculate the discriminant:
2. $$\Delta = (-7)^2 - 4(2)(3) = 49 - 24 = 25$$
3. $\Delta = 25 > 0$, so we have two distinct real roots. Apply the quadratic formula:
4. $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{7 \pm 5}{4}$$
5. Calculate the two roots: $x = \frac{12}{4} = 3$ and $x = \frac{2}{4} = 0.5$. The roots are $x=0.5$ and $x=3$.

> **Exam tip:** Always write down your discriminant calculation first — you get method marks for it even if you make a mistake in calculating the final roots.

## Finding the Vertex of a Quadratic

**Vertex of a parabola** — The turning point of the parabola. It is the minimum point if the parabola opens upwards ($a>0$) and the maximum point if it opens downwards ($a<0$). The x-coordinate of the vertex is always halfway between the two roots.

*Example:* For roots $x=1$ and $x=3$, the x-coordinate of the vertex is $\frac{1+3}{2} = 2$.

For a quadratic in standard form $ax^2 + bx + c$, the x-coordinate of the vertex can be calculated directly with the formula $x = -\frac{b}{2a}$. Substitute this value back into the function to get the y-coordinate.

**Worked example:** Find the vertex of $f(x) = 3x^2 - 12x + 7$, and state if it is a maximum or minimum.

1. Identify $a=3$, $b=-12$. Use the formula for x-coordinate of the vertex:
2. $$x = -\frac{b}{2a} = -\frac{(-12)}{2(3)} = 2$$
3. Substitute $x=2$ back into the function to find the y-coordinate:
4. $$f(2) = 3(2)^2 - 12(2) + 7 = 12 - 24 + 7 = -5$$
5. The vertex is at $(2, -5)$. Since $a=3 > 0$, the parabola opens upwards, so this is a minimum turning point.

> **Exam tip:** For optimization problems (maximum profit, minimum cost, maximum height), the answer is almost always the y-coordinate of the vertex.

## Interpreting Roots and Vertices in Context

Almost all quadratic questions in IB AI SL are context-based, so you need to interpret calculated values in the problem's scenario, not just give mathematical results.

**Worked example:** The height of a ball $t$ seconds after being thrown is $h(t) = -5t^2 + 20t + 2$ (height in meters). When does the ball hit the ground, and what is its maximum height?

1. The ball hits the ground when height is 0, so we need the positive root of $h(t)=0$ (negative time is impossible here). Maximum height is the y-coordinate of the vertex, since $a=-5 < 0$ (maximum vertex).
2. Calculate the vertex: time of maximum height is $t = -\frac{20}{2(-5)} = 2$ seconds. Maximum height is:
3. $$h(2) = -5(2)^2 + 20(2) + 2 = 22 \text{ meters}$$
4. Solve for the positive root using the quadratic formula:
5. $$t = \frac{20 \pm \sqrt{440}}{10} \approx 4.1 \text{ seconds (positive root)}$$
6. Interpretation: The ball reaches a maximum height of 22 m after 2 seconds, and hits the ground after approximately 4.1 seconds.

> **tip**
>
> Always check if your results make sense in context. Negative values for time, height, or length are almost never valid answers.

## Common pitfalls

- **Wrong:** Treating $0x^2 + 2x + 1$ as a quadratic and trying to apply the quadratic formula
  - Why it fails: If the coefficient of $x^2$ is zero, the function is linear, not quadratic, so it only has one root
  - Correct: Always confirm that the coefficient of $x^2$ is non-zero before using any quadratic methods
- **Wrong:** Mismanaging signs when calculating $x = -\frac{b}{2a}$, getting the wrong x-coordinate for the vertex
  - Why it fails: It is easy to forget the negative sign out front when $b$ is already negative
  - Correct: Write down $b$ and its sign explicitly before substitution, e.g. for $x^2 - 4x +3$, $b=-4$ so $x = -\frac{(-4)}{2(1)} = 2$
- **Wrong:** Leaving both positive and negative roots in a context problem like projectile motion
  - Why it fails: Examiners expect you to interpret results in context, not just output all mathematical solutions
  - Correct: Always discard any results that do not make sense in the problem's scenario before giving your final answer
- **Wrong:** Giving the x-coordinate of the vertex as the answer when asked for maximum height
  - Why it fails: It is easy to stop after calculating the x-coordinate, but questions usually ask for the maximum value, not the time it occurs at
  - Correct: Always check what the question asks for: maximum value = y-coordinate, time of maximum = x-coordinate

## Cheatsheet

| Feature | Standard Form: $ax^2+bx+c$ | Factored Form: $a(x-r_1)(x-r_2)$ | Vertex Form: $a(x-h)^2+k$ |
| --- | --- | --- | --- |
| Roots | $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ | Read as $r_1, r_2$ directly | Set equal to 0 and solve for $x$ |
| Number of real roots | $\Delta>0$: 2; $\Delta=0$: 1; $\Delta<0$: 0 | 2 distinct or 1 repeated | Same as standard form |
| Vertex x-coordinate | $-\frac{b}{2a}$ | $\frac{r_1 + r_2}{2}$ | Read as $h$ directly |
| Vertex y-coordinate | Substitute $x=-\frac{b}{2a}$ | Substitute $x=\frac{r_1+r_2}{2}$ | Read as $k$ directly |
| Y-intercept | Read as $c$ directly | Evaluate at $x=0$ | Evaluate at $x=0$ |
| Max/Min | Max if $a<0$, Min if $a>0$ | Same rule | Same rule |

## What's next

Quadratic functions are the foundation for more advanced non-linear modeling in IB AI SL, and they appear in almost every exam paper, often in 5-8 mark context-based questions. Understanding how to quickly identify and calculate roots and vertices allows you to solve common exam problems including optimization, projectile motion, and profit modeling efficiently. This sub-topic is also a prerequisite for working with other non-linear functions like exponential and cubic functions later in the course. Next, you will build on this knowledge to solve quadratic inequalities and use quadratic regression to model real-world bivariate data.

- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-exponential-and-logarithmic-functions/)
- [Transformations of function graphs](https://www.owlsprep.com/study/ib-math-ai-sl-u2-transformations-of-function-graphs/)
- [Real-world modeling with functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-real-world-modeling-with-functions/)

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