Quadratic Functions and Models: Precalculus 3.1 Study Notes

Quadratic Functions and Models

Quadratic Function in Standard Form

A quadratic function is a polynomial function of degree 2, typically written in standard form:

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

• Domain: All real numbers.

• Graph: The graph of a quadratic function is a parabola.

Example 1: Graph the function $f(x) = \frac{1}{2}x^2 - 3x + 2$.

• Analyze the coefficients $a$, $b$, and $c$ to determine the direction and position of the parabola.

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

Quadratic Function in Vertex Form

The vertex form of a quadratic function highlights the vertex of the parabola:

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

Example 2: Convert $f(x) = x^2 + 4x + 2$ to vertex form and graph the function.

• Complete the square to rewrite in vertex form.

• Find the vertex using $h = -\frac{b}{2a}$ and $k = f(h)$.

Properties of the Graph of a Quadratic Function

• Axis of Symmetry: $x = -\frac{b}{2a}$

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

• Direction: Parabola opens upward if $a > 0$, downward if $a < 0$.

• Range: $y \geq k$ if $a > 0$; $y \leq k$ if $a < 0$

Finding the Zeros (x-intercepts) of a Quadratic

The zeros of a quadratic function are the solutions to $f(x) = 0$.

• Also known as the x-intercepts or roots.

• Can be found by factoring or using the quadratic formula.

Methods for Solving Quadratic Equations

• Method 1: Factoring

  • Set $f(x) = 0$ and factor the quadratic expression.

  • Example: $f(x) = x^2 + x - 12$

• Method 2: Quadratic Formula

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

  • Example: $f(x) = 3x^2 - 4x + 1$

Key Features to Identify in a Quadratic Function

• Direction (Up/Down)

• Vertex

• Axis of Symmetry

• x-intercepts

• Range

The x-intercepts of a Quadratic Function

Condition	Number of x-intercepts	Description
$b^2 - 4ac > 0$	2	Two distinct real x-intercepts
$b^2 - 4ac = 0$	1	One real x-intercept (vertex touches x-axis)
$b^2 - 4ac < 0$	0	No real x-intercepts (parabola does not cross x-axis)

Writing the Equation of a Quadratic Given a Vertex and a Point

• Given vertex $(h, k)$ and a point $(x_1, y_1)$, use vertex form $f(x) = a(x-h)^2 + k$.

• Substitute $(x_1, y_1)$ to solve for $a$.

Applications of Quadratic Functions

• Projectile Motion: The height of a projectile as a function of time or horizontal distance is modeled by a quadratic equation.

• Example: $h(x) = -\frac{1}{32}x^2 + x + 500$ models the height of a projectile above water.

• Find maximum height (vertex) and when the projectile strikes the water (solve for $h(x) = 0$).

• Architecture: Parabolic arches can be modeled using quadratic equations.

• Example: An arch with a span of 120 ft and max height of 25 ft. Use coordinate axes to model and solve for height at specific points.

Data Modeling with Quadratic Functions

• Quadratic functions can fit data points, such as the height of an arch at various distances.

• Plot data, connect points, and find the quadratic function that best fits the data.

Summary Table: Quadratic Function Forms

Form	Equation	Key Features
Standard Form	$f(x) = ax^2 + bx + c$	Coefficients $a$, $b$, $c$; easy to expand and solve
Vertex Form	$f(x) = a(x-h)^2 + k$	Vertex $(h, k)$ is explicit; useful for graphing
Factored Form	$f(x) = a(x - r_1)(x - r_2)$	x-intercepts $r_1$, $r_2$ are explicit

Additional info:

• Quadratic functions are foundational for understanding polynomial behavior, graphing, and modeling real-world phenomena in precalculus.

• Homework assignments and real-life problems reinforce the application of quadratic models.