2.3 Quadratic Functions: Properties, Graphs, and Applications

Quadratic Functions

Introduction to the Square Function

The square function is one of the most fundamental elementary functions in algebra, defined as $h(x) = x^2$. Its graph is a U-shaped curve known as a parabola.

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

• All quadratic functions have graphs that are parabolas, which are similar in shape to the graph of $x^2$.

• Domain: The domain of any quadratic function is all real numbers, $(-\infty, \infty)$.

Example: The graph of $h(x) = x^2$ opens upward and is symmetric about the y-axis.

Key Features of Quadratic Functions

• Vertex: The highest or lowest point on the graph of a quadratic function. For $f(x) = ax^2 + bx + c$, the vertex occurs at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line that passes through the vertex, given by $x = -\frac{b}{2a}$.

• Direction: If $a > 0$, the parabola opens upward (minimum point). If $a < 0$, it opens downward (maximum point).

• Intercepts:

  • y-intercept: The point where the graph crosses the y-axis, found by evaluating $f(0) = c$.

  • x-intercepts (zeros): The points where the graph crosses the x-axis, found by solving $ax^2 + bx + c = 0$.

The Quadratic Formula

To find the x-intercepts (real solutions) of a quadratic equation $ax^2 + bx + c = 0$, use the quadratic formula:

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

The solutions are real if $b^2 - 4ac \geq 0$.

Forms of Quadratic Functions

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

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

Axis of symmetry: $x = h$ in vertex form.

Minimum/Maximum Value:

• If $a > 0$, the minimum value is $k$ at $x = h$.

• If $a < 0$, the maximum value is $k$ at $x = h$.

Graphing Quadratic Functions

To graph a quadratic function:

1. Identify the vertex using $x = -\frac{b}{2a}$ (standard form) or $(h, k)$ (vertex form).

2. Find the y-intercept by evaluating $f(0)$.

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

4. Plot the vertex, intercepts, and additional points as needed.

5. Draw the axis of symmetry and sketch the parabola.

Example: For $f(x) = -x^2 + 5x + 3$:

• Vertex: $x = -\frac{5}{2(-1)} = 2.5$

• y-intercept: $f(0) = 3$

• x-intercepts: Solve $-x^2 + 5x + 3 = 0$ using the quadratic formula.

Applications of Quadratic Functions

Quadratic functions are widely used in modeling real-world situations, such as projectile motion, profit maximization, and cost minimization.

• Profit and Revenue: Functions like $R(x)$ (revenue) and $C(x)$ (cost) are often quadratic. The difference $P(x) = R(x) - C(x)$ gives the profit function.

• Break-even Points: The values of $x$ where $R(x) = C(x)$ represent the break-even points.

• Optimization: The vertex of the profit or cost function gives the maximum profit or minimum cost, respectively.

Summary Table: Properties of Quadratic Functions

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

Practice and Application

• Given a quadratic function, be able to:

  • Find the domain

  • Identify the vertex

  • Determine the axis of symmetry

  • Find x- and y-intercepts

  • Determine the minimum or maximum value

  • Graph the function

  • State intervals of increase and decrease

• Apply quadratic functions to solve real-world problems involving revenue, cost, and profit.

Example Application: Suppose a company's revenue and cost functions are quadratic. To find the break-even point, set $R(x) = C(x)$ and solve for $x$. To maximize profit, find the vertex of the profit function $P(x) = R(x) - C(x)$.