Quadratic Functions: Concepts, Graphs, and Applications

Quadratic Functions

Introduction to Quadratic Functions

Quadratic functions are a fundamental class of polynomial functions that appear frequently in algebra and real-world applications. A quadratic function is any function that can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$ and $a$, $b$, and $c$ are real numbers.

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

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Axis of Symmetry: The vertical line $x = h$ (in vertex form) or $x = -\frac{b}{2a}$ (in standard form)

• Graph: The graph of a quadratic function is a parabola that opens upward if $a > 0$ and downward if $a < 0$.

Graphs of Quadratic Functions

The graph of a quadratic function is a parabola. The vertex is the highest or lowest point, depending on the direction the parabola opens. The axis of symmetry passes through the vertex and divides the parabola into two mirror images.

• Vertex: The point $(h, k)$ in vertex form or $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ in standard form.

• Direction: Determined by the sign of $a$.

• Y-intercept: The point $(0, c)$.

• X-intercepts (Roots): The solutions to $ax^2 + bx + c = 0$.

Graphing Quadratic Functions in Standard Form

To graph a quadratic function in standard form, follow these steps:

1. Find the vertex using $x = -\frac{b}{2a}$ and $y = f\left(-\frac{b}{2a}\right)$.

2. Determine the axis of symmetry ($x = -\frac{b}{2a}$).

3. Find the y-intercept ($f(0) = c$).

4. Find the x-intercepts by solving $ax^2 + bx + c = 0$ (if real solutions exist).

5. Plot additional points as needed for accuracy.

Graphing Quadratic Functions in Vertex Form

Vertex form makes it easy to identify the vertex and direction of the parabola. The function $f(x) = a(x-h)^2 + k$ has vertex $(h, k)$ and opens upward if $a > 0$, downward if $a < 0$.

• Plot the vertex $(h, k)$.

• Draw the axis of symmetry $x = h$.

• Choose points on either side of the vertex to plot the parabola.

Converting Between Forms

Quadratic functions can be converted between standard and vertex forms by completing the square:

• Given $f(x) = ax^2 + bx + c$, rewrite as $f(x) = a(x-h)^2 + k$ by completing the square.

• Vertex: $h = -\frac{b}{2a}$, $k = f(h)$.

Properties of Quadratic Functions

• Maximum/Minimum Value: The vertex gives the maximum (if $a < 0$) or minimum (if $a > 0$) value of the function.

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: $[k, \infty)$ if $a > 0$; $(-\infty, k]$ if $a < 0$.

Applications of Quadratic Functions

Quadratic functions model many real-world phenomena, such as projectile motion, area optimization, and revenue maximization. The vertex often represents the optimal value (maximum height, minimum cost, etc.).

• Projectile Motion: The height of an object thrown upward can be modeled by $h(t) = -16t^2 + vt + s$ (in feet, with $t$ in seconds).

• Optimization: Quadratic functions are used to find maximum or minimum values in various contexts.

Summary Table: Key Features of Quadratic Functions

Feature	Standard Form	Vertex Form
Equation	$f(x) = ax^2 + bx + c$	$f(x) = a(x-h)^2 + k$
Vertex	$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$	$(h, k)$
Axis of Symmetry	$x = -\frac{b}{2a}$	$x = h$
Y-intercept	$(0, c)$	$(0, a h^2 + k)$
Direction	Up if $a > 0$, Down if $a < 0$	Up if $a > 0$, Down if $a < 0$

Example: Finding the Vertex and Graphing

Given $f(x) = 2x^2 - 4x + 1$:

• Find the vertex: $x = -\frac{-4}{2 \times 2} = 1$, $y = f(1) = 2(1)^2 - 4(1) + 1 = -1$

• Vertex: $(1, -1)$

• Axis of symmetry: $x = 1$

• Y-intercept: $f(0) = 1$

• Graph the parabola opening upward.

Example: Application to Projectile Motion

A ball is thrown upward with an initial velocity of 32 ft/s from a height of 5 ft. Its height after $t$ seconds is $h(t) = -16t^2 + 32t + 5$.

• Maximum height occurs at $t = -\frac{32}{2 \times -16} = 1$ second.

• Maximum height: $h(1) = -16(1)^2 + 32(1) + 5 = 21$ ft.

Additional info: The notes also include step-by-step examples for graphing, converting forms, and solving application problems, as well as summary tables for quick reference.