D2-D4

Quadratic Functions

Definition and Standard Form

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

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

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

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

Example: $f(x) = x^2 - 2x - 3$ can be factored as $f(x) = (x-3)(x+1)$.

Key Features of Parabolas

• Vertex: The highest or lowest point on the graph. For $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line passing through the vertex, $x = -\frac{b}{2a}$.

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

• 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.

• Y-intercept: The point where $x = 0$, so $f(0) = c$.

• X-intercepts (Roots): The values of $x$ where $f(x) = 0$.

Example: For $f(x) = x^2 - 2x - 3$, the vertex is at $x = 1$, $y = -4$; axis of symmetry is $x = 1$; y-intercept is $(0, -3)$; x-intercepts are $(3, 0)$ and $(-1, 0)$.

Finding X-Intercepts of Quadratic Functions

To find the x-intercepts (roots) of $f(x) = ax^2 + bx + c$, solve $f(x) = 0$ using one of the following methods:

• Factoring: Express $f(x)$ as a product of linear factors and set each factor to zero.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $x^2 - 2x - 3 = 0$ by factoring: $(x-3)(x+1) = 0 \implies x = 3, -1$.

Example: Solve $x^2 + 2x + 1 = 0$ using the quadratic formula: $x = \frac{-2 \pm \sqrt{4 - 4}}{2} = -1$ (double root).

The Discriminant

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$.

• If $D > 0$, there are two distinct real roots (parabola crosses the x-axis twice).

• If $D = 0$, there is one real root (parabola is tangent to the x-axis).

• If $D < 0$, there are no real roots (parabola does not cross the x-axis).

Vertex Form and Transformations

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

• Horizontal shift: $h$ units right if $h > 0$, left if $h < 0$

• Vertical shift: $k$ units up if $k > 0$, down if $k < 0$

• Reflection: If $a < 0$, the parabola opens downward

• Vertical stretch/shrink: $|a| > 1$ stretches, $0 < |a| < 1$ shrinks

Example: $f(x) = 2(x-1)^2 + 3$ is a parabola opening upward, vertex at $(1, 3)$, vertically stretched by 2.

Applications of Quadratic Functions

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

• Projectile motion

• Maximizing/minimizing area or revenue

• Optimization problems

Example: The height $h$ of an object thrown upward from a platform can be modeled by $h(t) = -16t^2 + vt + h_0$.

• Find when the object hits the ground: set $h(t) = 0$ and solve for $t$.

• Find the maximum height: vertex at $t = -\frac{b}{2a}$.

Solving Application Problems

• Write the quadratic function that models the situation.

• Identify what is being maximized or minimized (e.g., area, height, profit).

• Find the vertex to determine the maximum or minimum value.

• Solve for intercepts as needed to answer the question.

Example: To maximize the area of a rectangular garden with a fixed perimeter, express area as a function of one variable, write in quadratic form, and find the vertex.

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	$x = -\frac{b}{2a}$
Y-intercept	$(0, c)$
X-intercepts	Solve $ax^2 + bx + c = 0$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant	$b^2 - 4ac$
Domain	$(-\infty, \infty)$
Range	Upward: $[k, \infty)$; Downward: $(-\infty, k]$

Practice Problems

• Find the vertex, axis of symmetry, domain, range, and intercepts of $f(x) = 2x^2 - 4x + 1$.

• Find the equation of a parabola with vertex $(2, -3)$ and passing through $(4, 5)$.

• Given $f(x) = -x^2 + 6x - 8$, find the maximum value and when it occurs.

• Application: A rectangular garden is to be fenced on three sides with 100 feet of fencing. What dimensions maximize the area?

Additional info: These notes cover all major aspects of quadratic functions, including their properties, graphing, solving, and applications, as required in a College Algebra course.