D2-D4

Quadratic Functions

Definition and General Form

A quadratic function is a polynomial function of degree 2, typically written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola.

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

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Factored Form: $f(x) = a(x-r_1)(x-r_2)$, where $r_1$ and $r_2$ are the roots

Example: $f(x) = 2x^2 - 3x + 1$

Graphing Quadratic Functions

The graph of a quadratic function is a parabola. The direction in which the parabola opens depends on the sign of $a$:

• If $a > 0$, the parabola opens upward.

• If $a < 0$, the parabola opens downward.

Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$ divides the parabola into two symmetric halves.

Vertex: The vertex is the highest or lowest point of the parabola, located at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.

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

Range: If $a > 0$, $[k, \infty)$; if $a < 0$, $(-\infty, k]$, where $k$ is the $y$-coordinate of the vertex.

Intercepts:

• y-intercept: $f(0) = c$

• x-intercepts: Solve $ax^2 + bx + c = 0$

Finding x-intercepts

To find the x-intercepts (roots or zeros) of a quadratic function, set $f(x) = 0$ and solve for $x$.

• Factoring: If possible, factor the quadratic and set each factor equal to zero.

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

Example: $x^2 - 5x + 6 = 0$ factors to $(x-2)(x-3) = 0$, so $x = 2$ and $x = 3$.

Discriminant and Number of Real Roots

The discriminant $D = b^2 - 4ac$ determines the number and type of roots:

• If $D > 0$, two distinct real roots.

• If $D = 0$, one real root (a repeated root).

• If $D < 0$, no real roots (two complex roots).

Example: For $x^2 + 4x + 5 = 0$, $D = 16 - 20 = -4$ (no real roots).

Vertex Form and Transformations

The vertex form $f(x) = a(x-h)^2 + k$ is useful for identifying the vertex and describing transformations:

• Vertex: $(h, k)$

• Horizontal shift: $h$

• Vertical shift: $k$

• Stretch/Compression: $|a|$

• Reflection: If $a < 0$, the parabola is reflected over the $x$-axis.

Example: $f(x) = 2(x-3)^2 + 4$ has vertex $(3, 4)$ and opens upward.

Applications of Quadratic Functions

Quadratic functions are used to model many real-world situations:

• Projectile motion: Height as a function of time

• Maximizing area: Optimization problems

• Minimizing cost: Economics applications

Example: The height $h$ of an object thrown upward: $h(t) = -16t^2 + vt + h_0$

Solving Application Problems

To solve application problems involving quadratics:

1. Write the quadratic equation modeling the situation.

2. Identify the vertex, intercepts, and other relevant features.

3. Interpret the solution in the context of the problem.

Example: Maximizing the area of a garden with a fixed perimeter.

Summary Table: Characteristics of Quadratic Functions

Feature	Formula/Description
Standard Form	$f(x) = ax^2 + bx + c$
Vertex	$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$
Axis of Symmetry	$x = -\frac{b}{2a}$
Discriminant	$D = b^2 - 4ac$
x-intercepts	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
y-intercept	$f(0) = c$
Domain	$(-\infty, \infty)$
Range	If $a > 0$, $[k, \infty)$; if $a < 0$, $(-\infty, k]$

Practice Problems

• Find the vertex, axis of symmetry, domain, range, and intercepts of $f(x) = 2x^2 - 8x + 5$.

• Find the equation of a parabola with vertex $(2, -4)$ passing through $(4, 0)$.

• Maximize the area of a rectangular garden with a fixed perimeter using quadratic modeling.

Additional info: These notes cover the topic of quadratic functions, including their properties, graphing techniques, solving equations, and applications, which are central to College Algebra (Chapter 4: Polynomial Functions).