Quadratic Functions: Graphing, Properties, and Applications

Quadratic Functions

Introduction to Quadratic Functions

Quadratic functions are a fundamental topic in precalculus, representing polynomial functions of degree two. Their graphs are parabolas, which can open upward or downward depending on the leading coefficient.

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

• Parabola: The graph of a quadratic function, which is symmetric about a vertical line called the axis of symmetry.

Standard (Vertex) Form of a Quadratic Function

The vertex form of a quadratic function highlights the vertex and orientation of the parabola.

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

• Axis of symmetry: The line $x = h$.

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

Graphing Quadratic Functions in Vertex Form

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

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

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

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

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

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

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

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

• Step 2: Vertex is $(1, 4)$.

• Step 3: Find x-intercepts: $0 = - (x - 1)^2 + 4$ $-4 = - (x - 1)^2$ $4 = (x - 1)^2$ $x - 1 = ext{±}2$ $x = 3$ or $x = -1$ x-intercepts: $(-1, 0)$ and $(3, 0)$.

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

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

Key Properties of Quadratic Functions

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

• Axis of symmetry: Vertical line passing through the vertex.

• Intercepts: Points where the graph crosses the axes.

Applications of Quadratic Functions

Quadratic functions are used to model various real-world phenomena, such as projectile motion, area optimization, and economics.

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

• Optimization: Quadratic functions are often used to find optimal values in problems involving area, profit, or other quantities.

Summary Table: Quadratic Function Properties

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

Example Applications

• Finding the vertex: For $f(x) = ax^2 + bx + c$, vertex at $x = -\frac{b}{2a}$.

• Maximum/Minimum value: Substitute $x = -\frac{b}{2a}$ into $f(x)$ to find the value.

• Optimization problem: Maximizing area with a fixed perimeter, as in fencing problems.

Additional info: These notes are based on textbook slides and cover the essential concepts and procedures for graphing and analyzing quadratic functions in precalculus.