Quadratic Functions and Applications in College Algebra

Quadratic Functions

Definition and Standard Form

Quadratic functions are polynomial functions of degree two, commonly used to model various real-world phenomena. The general form of a quadratic function is:

• Standard form: $f(x) = ax^2 + bx + c$, where a, b, and c are real numbers and $a \neq 0$.

• Vertex form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

The graph of a quadratic function is a parabola, which opens upward if $a > 0$ and downward if $a < 0$.

Key Properties of Quadratic Functions

• Vertex: The vertex is the turning point of the parabola. For the standard form, the vertex $(h, k)$ can be found using:

  • $h = -\frac{b}{2a}$

  • $k = f\left(-\frac{b}{2a}\right)$

• Axis of Symmetry: The vertical line $x = h$ passes through the vertex and divides the parabola into two symmetric halves.

• Minimum/Maximum Value:

  • If $a > 0$, the vertex is a minimum point; the minimum value is $k$.

  • If $a < 0$, the vertex is a maximum point; the maximum value is $k$.

• Quadratic Formula: The solutions (roots) of $ax^2 + bx + c = 0$ are given by:

  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Identifying Function Types

Determine whether the following functions are linear, constant, quadratic, or neither:

• A) $y = x(x - 9) + 3x + 5$ — Quadratic (expands to $x^2 - 6x + 5$)

• B) $y = \frac{2x - 9}{3}$ — Linear

• C) $y = \frac{3x + 5}{2x}$ — Neither (rational function)

• D) $y = 4x - 3(x + 2) - x$ — Linear (simplifies to $4x - 3x - 6 - x = -6$; Constant)

Graphing Quadratic Functions

Key Features to Identify

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

• Axis of Symmetry: $x = h$.

• Y-intercept: Set $x = 0$ in $f(x)$.

• X-intercepts: Solve $f(x) = 0$ using the quadratic formula.

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

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

Graphs are typically plotted on a Cartesian plane, showing the vertex, axis of symmetry, and intercepts.

Example Functions for Graphing

• b) $f(x) = x^2 + 2x - 4$

• c) $f(x) = 3x^2 - 2x + 5$

• d) $g(x) = -x^2 + 18x + 19$

For each function, calculate the vertex, axis of symmetry, intercepts, and sketch the graph.

Applications of Quadratic Functions

Break-Even Analysis

Quadratic functions are often used in business to model revenue, cost, and profit. Consider:

• Revenue function: $R(x) = x(200 - 4x)$

• Cost function: $C(x) = 160 + 20x$

Where $x$ is the number of components (in millions).

• Break-even point: Solve $R(x) = C(x)$ for $x$.

• Profit function: $P(x) = R(x) - C(x)$

• Maximum profit: Find the vertex of $P(x)$.

Example: Cost Optimization

Given a cost function for building a house:

• $C(x) = 0.01x^2 - 20x + 25,000$ where $1000 \leq x \leq 3500$

• a) Find the cost for $x = 1500$ square feet.

• b) Find $x$ if $C(x) = 25,000$.

• c) Find the minimum cost and corresponding $x$ (vertex).

Summary Table: Quadratic Function Properties

Property	Description	Formula
Standard Form	General quadratic equation	$f(x) = ax^2 + bx + c$
Vertex	Turning point of the parabola	$h = -\frac{b}{2a}$, $k = f(h)$
Axis of Symmetry	Vertical line through vertex	$x = h$
Quadratic Formula	Roots of the quadratic equation	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Y-intercept	Value at $x = 0$	$f(0) = c$
X-intercepts	Where $f(x) = 0$	Use quadratic formula

Additional info:

• Quadratic functions are fundamental in algebra and appear in physics, engineering, economics, and many other fields.

• Understanding how to graph and analyze quadratic functions is essential for solving optimization and modeling problems.