Quadratic and Polynomial Functions: Concepts, Graphs, and Applications

Section 4.1 Quadratic Functions

4.1.1 Solving Quadratic Equations

Quadratic equations can be solved using several methods, including factoring, completing the square, and the quadratic formula. Each method is useful in different contexts and provides insight into the properties of quadratic functions.

• Factoring and the Zero Product Property: To solve $5x^2 + 14x - 3 = 0$ by factoring, rewrite the equation in factored form and set each factor equal to zero. The solutions are $x = -3$ or $x = \frac{1}{5}$.

• Completing the Square: To complete the square for $x^2 + \frac{5}{6}x$, add $\frac{25}{144}$ to form a perfect square trinomial.

• Quadratic Formula: For $2x^2 - 2x - 3 = 0$, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find $x = \frac{1 + \sqrt{25}}{2}$ or $x = \frac{1 - \sqrt{25}}{2}$.

Example: Solve $x^2 + 5x + 6 = 0$ by factoring: $(x+2)(x+3) = 0 \implies x = -2, -3$.

4.1.2 Domain and Range from Graphs

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). For quadratic functions, the domain is typically all real numbers, while the range depends on the vertex and the direction the parabola opens.

• To determine the domain and range from a graph, identify the leftmost and rightmost points (domain) and the lowest or highest point (range).

4.1.3 Transformations of Quadratic Functions

Quadratic functions can be transformed by shifting, stretching, compressing, or reflecting their graphs. The general form is $f(x) = a(x-h)^2 + k$.

• Vertical Shifts: Adding or subtracting a constant $k$ moves the graph up or down.

• Horizontal Shifts: Changing $h$ moves the graph left or right.

• Reflections: A negative $a$ reflects the graph across the x-axis.

• Stretching/Compressing: The value of $|a|$ stretches ($|a| > 1$) or compresses ($0 < |a| < 1$) the graph vertically.

Example: $f(x) = -2(x+3)^2 - 1$ is a parabola opening down, shifted left 3 units and down 1 unit, and vertically stretched by a factor of 2.

4.1.4 Definition and Properties of Quadratic Functions

A quadratic function is a polynomial function of degree 2, generally written as $f(x) = ax^2 + bx + c$, where $a \neq 0$.

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

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

Key Characteristics of a Parabola:

1. Vertex: The highest or lowest point of the parabola, given by $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$.

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

3. y-intercept: The point where the graph crosses the y-axis, at $(0, c)$.

4. x-intercepts (real zeros): The points where the graph crosses the x-axis, found by solving $ax^2 + bx + c = 0$.

5. Domain and Range: Domain is $(-\infty, \infty)$. Range depends on the vertex and direction of opening.

4.1.5 Vertex Form of a Quadratic Function

The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

• The graph opens up if $a > 0$ and down if $a < 0$.

• Axis of symmetry: $x = h$.

• y-intercept: Substitute $x = 0$ into the equation.

• x-intercepts: Set $f(x) = 0$ and solve for $x$.

Example: For $f(x) = -(x-2)^2 - 4$, the vertex is $(2, -4)$, the graph opens down, and the axis of symmetry is $x = 2$.

4.1.6 Completing the Square

Completing the square is a method to rewrite a quadratic function in vertex form and to solve quadratic equations.

• Rewrite $ax^2 + bx + c$ as $a(x-h)^2 + k$ by completing the square.

• Helps identify the vertex and graph the function easily.

Example: Rewrite $f(x) = 2x^2 - 4x - 3$ in vertex form by completing the square.

4.1.7 Formula for the Vertex of a Parabola

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

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

4.1.8 Maximum and Minimum Values

The vertex of a parabola represents the maximum (if $a < 0$) or minimum (if $a > 0$) value of the quadratic function.

• Maximum or minimum value is the y-coordinate of the vertex.

4.1.9 Determining the Equation from a Graph

Given a graph, you can determine the equation of a quadratic function by identifying the vertex, axis of symmetry, and another point to solve for $a$.

• Vertex form: $f(x) = a(x-h)^2 + k$

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

Section 4.2 Applications and Modeling of Quadratic Functions

4.2.1 Projectile Motion Model

Quadratic functions are used to model projectile motion, where the height of an object is a function of time or horizontal distance.

• General form: $h(t) = -\frac{1}{2}gt^2 + v_0 t + h_0$, where $g$ is the acceleration due to gravity, $v_0$ is initial velocity, and $h_0$ is initial height.

• Maximum height occurs at the vertex.

Example: $h(t) = -4.9t^2 + 44t + 1$ models the height of a toy rocket.

4.2.2 Revenue, Demand, and Profit Models

Quadratic functions can model business applications such as revenue, demand, and profit.

• Revenue: $R = p \cdot x$, where $p$ is price and $x$ is quantity sold.

• Profit: $P = R - C$, where $C$ is the cost function.

• Maximum revenue or profit occurs at the vertex of the quadratic function.

Example: If $p = 10 - \frac{1}{50}x$, then $R(x) = x(10 - \frac{1}{50}x)$.

4.2.3 Optimization Problems

Quadratic functions are used to solve optimization problems, such as maximizing area or profit under given constraints.

• Express the quantity to be maximized as a quadratic function of one variable.

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

Example: To maximize the area of a rectangular fence with a fixed perimeter, write area as a function of one side and find its maximum.

Section 4.3 The Graphs of Polynomial Functions

4.3.1 Intercepts of a Function

Polynomial functions can have multiple x- and y-intercepts, depending on their degree.

• x-intercepts: Solve $f(x) = 0$.

• y-intercept: Evaluate $f(0)$.

Example: For $f(x) = x^3 - 2x^2 + x - 2$, find all intercepts by solving for $x$ and evaluating at $x = 0$.

4.3.2 Graphs of Basic Functions

Understanding the basic shapes of polynomial functions helps in graphing more complex functions.

• Identity function: $f(x) = x$ (straight line through the origin).

• Square function: $f(x) = x^2$ (parabola opening up).

• Cubic function: $f(x) = x^3$ (S-shaped curve).

4.3.3 Definition and Properties of Polynomial Functions

A polynomial function is a function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a non-negative integer and coefficients $a_i$ are real numbers.

• Degree: The highest power of $x$ with a nonzero coefficient.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term without $x$.

Example: $f(x) = 4x^3 - 3x^2 + x - 3$ is a cubic polynomial with degree 3, leading coefficient 4, and constant term -3.

4.3.4 Identifying Polynomial Functions

Not all functions with $x$ are polynomials. A function is a polynomial if all exponents are non-negative integers and all coefficients are real numbers.

• Example: $f(x) = \sqrt{x^5} - 2x^2 - \frac{1}{6}$ is not a polynomial because of the square root.

• Example: $g(x) = 4x^4 - 3x^2 + x^3 + \frac{7}{x} - 3$ is not a polynomial because of the $\frac{7}{x}$ term.

• Example: $h(x) = \frac{3x - x^4 + 7x^4}{9}$ is a polynomial after simplification.

4.3.5 Summary Table: Key Properties of Quadratic and Polynomial Functions

Property	Quadratic Function	Polynomial Function
General Form	$ax^2 + bx + c$	$a_nx^n + \ldots + a_0$
Degree	2	Any non-negative integer
Graph Shape	Parabola	Varies (lines, parabolas, cubic curves, etc.)
Vertex	Yes	Only for degree 2
Axis of Symmetry	Yes	Only for even degree, symmetric polynomials
Intercepts	Up to 2 x-intercepts, 1 y-intercept	Up to n x-intercepts, 1 y-intercept