Chapter 2: Quadratic Functions & Equations

Quadratic functions are among the most important and widely applied functions in all of mathematics. From the trajectory of a thrown ball to the shape of a satellite dish, parabolas appear everywhere in the natural and engineered world. In this chapter, we develop a complete toolkit for understanding, graphing, solving, and applying quadratic functions and equations. By the end, you will be able to convert fluently between forms, solve any quadratic equation using multiple methods, analyze the discriminant, solve quadratic inequalities, and model real-world situations with quadratics.

2.1 Quadratic Functions & Parabolas

A quadratic function is any function that can be written in the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The requirement that $a \neq 0$ is essential: if $a = 0$, the $x^2$ term vanishes and we are left with a linear function. The graph of every quadratic function is a U-shaped curve called a parabola.

The sign of $a$ controls the direction the parabola opens. When $a > 0$, the parabola opens upward and has a minimum value at its vertex. When $a < 0$, the parabola opens downward and has a maximum value at its vertex. The magnitude $|a|$ controls the width: larger values of $|a|$ produce a narrower parabola, while smaller values produce a wider one.

Vertex Form

While standard form is convenient for identifying the $y$-intercept and for algebraic manipulation, vertex form is far more useful for graphing and understanding the geometry of a parabola.

Every quadratic function has exactly one vertex. The vertical line passing through the vertex, $x = h$, is called the axis of symmetry. The parabola is a mirror image of itself on either side of this line. For any point $(x_1, y_1)$ on the parabola, the point $(2h - x_1, y_1)$ is also on the parabola.

Converting Standard Form to Vertex Form

Given $f(x) = ax^2 + bx + c$, the vertex is located at:

Therefore, the vertex form is $f(x) = a\!\left(x + \dfrac{b}{2a}\right)^2 + \left(c - \dfrac{b^2}{4a}\right)$. In practice, it is often easier to find $h$ first using $h = -\frac{b}{2a}$ and then compute $k = f(h)$ directly.

Converting Vertex Form to Standard Form

To convert from vertex form to standard form, simply expand the squared binomial and simplify. For example, $f(x) = 3(x - 2)^2 + 5 = 3(x^2 - 4x + 4) + 5 = 3x^2 - 12x + 12 + 5 = 3x^2 - 12x + 17$.

In the interactive graph above, the blue parabola shows $y = a(x-h)^2 + k$ and the dashed gray curve shows the parent function $y = x^2$ for reference. Experiment with the sliders to observe:

• Changing $a$ from positive to negative flips the parabola upside down.
• Increasing $|a|$ makes the parabola narrower; decreasing $|a|$ makes it wider.
• Moving $h$ shifts the parabola left and right; the vertex follows along.
• Moving $k$ shifts the parabola up and down.

2.2 Factoring Quadratics

Factoring is the process of rewriting a quadratic expression as a product of simpler expressions. It is one of the most fundamental algebraic skills and provides the fastest route to finding the roots (zeros) of many quadratic equations. When we factor $ax^2 + bx + c = 0$ into a product of linear factors, we can immediately apply the Zero Product Property: if $AB = 0$, then $A = 0$ or $B = 0$.

Greatest Common Factor (GCF)

Before applying any other factoring technique, always look for a greatest common factor first. If every term shares a common factor, pull it out.

Factoring Trinomials ($a = 1$)

For a trinomial of the form $x^2 + bx + c$, we seek two numbers $p$ and $q$ such that $p + q = b$ and $pq = c$. If such numbers exist, then $x^2 + bx + c = (x + p)(x + q)$.

Factoring Trinomials ($a \neq 1$): The AC Method

When the leading coefficient is not $1$, we use the AC method (also called factoring by grouping). For $ax^2 + bx + c$:

1. Compute the product $ac$.
2. Find two numbers $p$ and $q$ such that $p + q = b$ and $pq = ac$.
3. Rewrite the middle term $bx$ as $px + qx$.
4. Factor by grouping.

Difference of Squares

Perfect Square Trinomials

2.3 Completing the Square

Completing the square is a powerful algebraic technique that transforms a quadratic expression into vertex form. It is useful for three purposes: converting a quadratic to vertex form for graphing, solving quadratic equations, and deriving the quadratic formula itself.

The Method

To complete the square for $ax^2 + bx + c$:

1. If $a \neq 1$, factor out $a$ from the first two terms: $a\!\left(x^2 + \frac{b}{a}x\right) + c$.
2. Take half the coefficient of $x$ inside the parentheses and square it: $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$.
3. Add and subtract this value inside the parentheses to create a perfect square trinomial.
4. Factor the perfect square trinomial and simplify the constant terms.

Solving Equations by Completing the Square

We can solve any quadratic equation $ax^2 + bx + c = 0$ by completing the square and then using the square root property: if $(x - h)^2 = d$, then $x - h = \pm\sqrt{d}$, so $x = h \pm \sqrt{d}$.

2.4 The Quadratic Formula & Discriminant

The quadratic formula provides a universal method for solving any quadratic equation. We derive it by completing the square on the general equation $ax^2 + bx + c = 0$.

Derivation of the Quadratic Formula

Starting with $ax^2 + bx + c = 0$ where $a \neq 0$:

Complete the square by adding $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$ to both sides:

The Discriminant

The expression under the square root, $\Delta = b^2 - 4ac$, is called the discriminant. It determines the nature and number of solutions without requiring us to solve the equation completely.

In the interactive graph above, the parabola $y = ax^2 + bx + c$ is shown in blue. The discriminant value $\Delta = b^2 - 4ac$ is displayed. As you adjust the sliders, notice:

• When $\Delta > 0$, the parabola crosses the $x$-axis at two points (two real roots).
• When $\Delta = 0$, the parabola just touches the $x$-axis at one point (one repeated root).
• When $\Delta < 0$, the parabola floats entirely above or below the $x$-axis (no real roots).

2.5 Quadratic Inequalities

A quadratic inequality has the form $ax^2 + bx + c > 0$, $ax^2 + bx + c < 0$, $ax^2 + bx + c \geq 0$, or $ax^2 + bx + c \leq 0$. To solve these, we combine our knowledge of factoring (or the quadratic formula) with an analysis of where the parabola lies above or below the $x$-axis.

Solving Quadratic Inequalities Algebraically

The general strategy is:

1. Move all terms to one side so that the inequality compares a quadratic expression to zero.
2. Find the roots of the corresponding equation $ax^2 + bx + c = 0$ (by factoring or the quadratic formula).
3. Use the roots to divide the number line into intervals.
4. Test a value from each interval to determine the sign of the quadratic on that interval.
5. Select the intervals that satisfy the inequality, and include or exclude endpoints depending on whether the inequality is strict ($<$, $>$) or non-strict ($\leq$, $\geq$).

Sign Charts

A sign chart is a compact visual representation of the analysis above. We write the factors along the left side, mark the critical values (roots) on a number line, and record the sign of each factor in each interval. Multiplying the signs gives the overall sign of the expression.

Graphical Interpretation

Graphically, solving $ax^2 + bx + c > 0$ asks: for which $x$-values is the parabola above the $x$-axis? Solving $ax^2 + bx + c < 0$ asks: for which $x$-values is the parabola below the $x$-axis? The roots mark the boundary between these regions. If the parabola opens upward ($a > 0$), it is below the $x$-axis between the roots and above elsewhere. If it opens downward ($a < 0$), the pattern reverses.

2.6 Applications of Quadratics

Quadratic functions model a remarkable variety of real-world situations. In this section, we explore three major application areas: projectile motion, optimization, and geometric problems.

Projectile Motion

When an object is launched near the surface of the Earth (ignoring air resistance), its height $h$ at time $t$ follows a quadratic model:

where $g \approx 9.8$ m/s$^2$ (or $32$ ft/s$^2$) is the acceleration due to gravity, $v_0$ is the initial upward velocity, and $h_0$ is the initial height. Since the leading coefficient is negative, the parabola opens downward, and the vertex gives the maximum height.

Optimization Problems

Since a quadratic function has either a maximum or a minimum at its vertex, we can use quadratics to find optimal values. If we can express the quantity to be optimized as a quadratic function of a single variable, we simply find the vertex.

Area Problems

2.7 Practice Problems

Test your understanding of the concepts in this chapter. Each problem has a detailed solution that you can reveal after attempting it yourself. Try each problem on paper before checking the answer.