Quadratic Functions: Properties, Graphing, and Applications

Quadratic Functions

Introduction to Quadratic Functions

Quadratic functions are a fundamental class of polynomial functions that play a central role in precalculus. Their graphs are parabolas, and they are widely used in modeling real-world phenomena involving maximum and minimum values.

• Quadratic function: A function of the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

• Parabola: The graph of a quadratic function, which is symmetric and has a distinct vertex (maximum or minimum point).

Standard and Vertex Forms of Quadratic Functions

Vertex Form

The vertex form of a quadratic function is especially useful for graphing and identifying the vertex of the parabola.

• Vertex form: $f(x) = a(x - h)^2 + k$, where $a \neq 0$.

• The vertex of the parabola is at the point $(h, k)$.

• The axis of symmetry is the vertical line $x = h$.

• If $a > 0$, the parabola opens upward, if $a < 0$, it opens downward.

Standard Form

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. The vertex can be found using formulas derived from this form.

• Vertex (x-coordinate): $x = -\frac{b}{2a}$

• Vertex (y-coordinate): $y = f\left(-\frac{b}{2a}\right)$

Graphing Quadratic Functions

Steps for Graphing in Vertex Form

To graph $f(x) = a(x - h)^2 + k$:

1. Determine the direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

2. Find the vertex: The vertex is $(h, k)$.

3. Find the x-intercepts: Solve $f(x) = 0$ for $x$.

4. Find the y-intercept: Compute $f(0)$.

5. Plot points: Plot the vertex, intercepts, and additional points as needed. Connect with a smooth, symmetric curve.

Example: Graphing in Vertex Form

Graph $f(x) = - (x - 1)^2 + 4$.

• Step 1: $a = -1$, so the parabola opens downward.

• Step 2: Vertex is at $(1, 4)$.

• Step 3: Find x-intercepts by solving $0 = - (x - 1)^2 + 4$:

  • $- (x - 1)^2 + 4 = 0$

  • $(x - 1)^2 = 4$

  • $x - 1 = \pm 2$

  • $x = 3$ or $x = -1$

• Step 4: Find y-intercept: $f(0) = - (0 - 1)^2 + 4 = -1 + 4 = 3$

• Summary: The parabola opens downward, has vertex $(1, 4)$, x-intercepts at $x = -1$ and $x = 3$, and y-intercept at $y = 3$.

Key Properties of Quadratic Functions

Vertex, Axis of Symmetry, and Intercepts

• Vertex: The highest or lowest point of the parabola, depending on the sign of $a$.

• Axis of symmetry: The vertical line passing through the vertex, $x = h$ (vertex form) or $x = -\frac{b}{2a}$ (standard form).

• x-intercepts: Points where $f(x) = 0$; solve the quadratic equation for $x$.

• y-intercept: The point where $x = 0$; $f(0) = c$ in standard form.

Maximum and Minimum Values

Determining Extrema

The vertex of a quadratic function represents its maximum or minimum value.

• If $a > 0$, the parabola opens upward and the vertex is a minimum.

• If $a < 0$, the parabola opens downward and the vertex is a maximum.

• The x-coordinate of the vertex is $x = -\frac{b}{2a}$.

• The minimum or maximum value is $f\left(-\frac{b}{2a}\right)$.

Example: Finding Maximum Area

Problem: You have 120 feet of fencing to enclose a rectangular region. What dimensions maximize the area?

1. Let $x$ = length, $y$ = width. Perimeter: $2x + 2y = 120 \implies y = 60 - x$

2. Area function: $A(x) = x(60 - x) = 60x - x^2$

3. Maximum area at vertex: $x = -\frac{b}{2a} = -\frac{60}{2 \times (-1)} = 30$

4. Dimensions: $x = 30$ ft, $y = 60 - 30 = 30$ ft

5. Maximum area: $A = 30 \times 30 = 900$ square feet

Summary Table: Forms and Properties of Quadratic Functions

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

Applications of Quadratic Functions

• Modeling projectile motion

• Maximizing or minimizing area, profit, or other quantities

• Solving real-world problems involving parabolic shapes