Quadratic Functions: Concepts, Graphing, and Applications

Section 4.1: Quadratic Functions

Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

• Factoring and the Zero Product Property: If a quadratic can be factored, set each factor equal to zero and solve for x. Example: Solve $x^2 + 5x + 6 = 0$ Factoring: $(x + 2)(x + 3) = 0$ Solutions: $x = -2,\ x = -3$

• Completing the Square: Rewrite the equation in the form $(x + p)^2 = q$ and solve for x. Example: $x^2 + 6x + 5 = 0$ Move constant: $x^2 + 6x = -5$ Add $(\frac{6}{2})^2 = 9$ to both sides: $x^2 + 6x + 9 = 4$ So, $(x + 3)^2 = 4$ Solutions: $x + 3 = \pm 2 \implies x = -1,\ x = -5$

• Quadratic Formula: For any quadratic $ax^2 + bx + c = 0$, the solutions are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Graphing Quadratic Functions

A quadratic function is any function that can be written as $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola.

• 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).

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

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

Characteristics of a Parabola

• Vertex: The highest or lowest point of the parabola.

• Axis of Symmetry: The vertical line passing through the vertex.

• Y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• X-intercepts (Roots): The points where the graph crosses the x-axis (solutions to $f(x) = 0$).

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

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

Objective 1: Understanding the Definition of a Quadratic Function and Its Graph

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph is a parabola that opens upward if $a > 0$ and downward if $a < 0$.

• Leading Coefficient: Determines the direction the parabola opens.

• Sketching: Draw the axis of symmetry, vertex, and a few points on either side.

Objective 2: Graphing Quadratic Functions Written in Vertex Form

The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$. This form makes it easy to identify the vertex $(h, k)$ and the direction the parabola opens.

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• Y-intercept: Set $x = 0$ and solve for $f(0)$

• Graphing Steps:

  1. Plot the vertex $(h, k)$

  2. Draw the axis of symmetry

  3. Find and plot the y-intercept and additional points

  4. Sketch the parabola

Objective 3: Graphing Quadratic Functions by Completing the Square

Completing the square is a method to rewrite a quadratic function in vertex form, which helps in graphing and identifying the vertex.

• Given $f(x) = ax^2 + bx + c$, factor out $a$ from the $x^2$ and $x$ terms, then complete the square inside the parentheses.

• Rewrite as $f(x) = a(x - h)^2 + k$ to identify the vertex and graph.

Objective 4: Graphing Quadratic Functions Using the Vertex Formula

The vertex of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the vertex formula:

• Vertex Formula: $x = -\frac{b}{2a}$

• Substitute $x$ into $f(x)$ to find the $y$-coordinate of the vertex.

• Use the vertex, axis of symmetry, and intercepts to sketch the graph.

Objective 5: Determining the Equation of a Quadratic Function Given Its Graph

Given a graph of a parabola, you can determine the equation by identifying the vertex, axis of symmetry, and another point on the graph.

• Determine if the parabola opens up or down (sign of $a$).

• Identify the vertex $(h, k)$ and write the equation in vertex form: $f(x) = a(x - h)^2 + k$.

• Use another point $(x_1, y_1)$ to solve for $a$.

• Rewrite in standard form if needed: $f(x) = ax^2 + bx + c$.

Summary Table: Key Features of Quadratic Functions

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

Additional info: These notes cover the core concepts of quadratic functions, including solving, graphing, and analyzing their properties, as required in a college algebra course.