Quadratic and Polynomial Functions: Definitions, Properties, and Theorems

Quadratic Functions and Their Graphs

Definition and Forms

A quadratic function is a function of the form:

• General form: $f(x) = ax^2 + bx + c$, $a \neq 0$

• Vertex form: $f(x) = a(x-h)^2 + k$, $a \neq 0$

The graph of a quadratic function is called a parabola. Every parabola has a vertical line called the axis of symmetry, which passes through the vertex (the highest or lowest point of the parabola).

Properties of Quadratic Functions

• Vertex (vertex form): For $f(x) = a(x-h)^2 + k$, the vertex is $(h, k)$ and the axis of symmetry is $x = h$.

• Vertex (general form): For $f(x) = ax^2 + bx + c$, the vertex is $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ and the axis of symmetry is $x = -\frac{b}{2a}$.

• Direction: If $a > 0$, the parabola opens upward (vertex is a minimum). If $a < 0$, the parabola opens downward (vertex is a maximum).

Examples

• Example 1: $f(x) = x^2 + 6x + 5$

  • Opens upward ($a = 1 > 0$)

  • Vertex: $x = -\frac{6}{2 \cdot 1} = -3$, $y = f(-3) = (-3)^2 + 6(-3) + 5 = 9 - 18 + 5 = -4$; vertex is $(-3, -4)$

  • Axis of symmetry: $x = -3$

  • Intercepts: $y$-intercept at $x=0$, $f(0) = 5$; $x$-intercepts by solving $x^2 + 6x + 5 = 0$

  • Vertex is a minimum

  

• Example 2: $f(x) = -x^2 - 4x - 3$

  • Opens downward ($a = -1 < 0$)

  • Vertex: $x = -\frac{-4}{2 \cdot -1} = -2$, $y = f(-2) = -(-2)^2 - 4(-2) - 3 = -4 + 8 - 3 = 1$; vertex is $(-2, 1)$

  • Axis of symmetry: $x = -2$

  • Vertex is a maximum

  

• Example 3: $f(x) = 2(x-1)^2 - 2$

  • Opens upward ($a = 2 > 0$)

  • Vertex: $(1, -2)$

  • Axis of symmetry: $x = 1$

  • Vertex is a minimum

  

Converting to Vertex Form

• Complete the square to rewrite $f(x) = ax^2 + bx + c$ as $f(x) = a(x-h)^2 + k$.

Applications

• Projectile motion: The height of a thrown object can be modeled by a quadratic function, e.g., $s(t) = -16t^2 + 64t + 100$.

• Optimization: Maximizing area or other quantities often leads to quadratic functions.

Polynomial Functions and Their Graphs

Definition

A polynomial function of degree $n$ is defined as:

• $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$

• Leading coefficient: $a_n$

End Behaviors of Polynomial Functions

• Leading Coefficient Test: The end behavior of the graph depends on the sign of the leading coefficient and the degree $n$.

• If $n$ is even and $a_n > 0$, both ends rise; if $a_n < 0$, both ends fall.

• If $n$ is odd and $a_n > 0$, left end falls and right end rises; if $a_n < 0$, left end rises and right end falls.

Zeros and Multiplicities

• Zero: $c$ is a zero of $f(x)$ if $f(c) = 0$.

• Multiplicity: If $(x-c)^m$ is a factor, $c$ is a zero of multiplicity $m$.

• If $m$ is odd, the graph crosses the $x$-axis at $c$; if $m$ is even, the graph touches and turns around at $c$.

Turning Points

• A polynomial of degree $n$ has at most $n-1$ turning points.

Intermediate Value Theorem (IVT)

• If $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$.

Dividing Polynomials, Remainder and Factor Theorems

Long Division

• For polynomials $f(x)$ and $d(x)$, $f(x) = d(x)q(x) + r(x)$, where $r(x)$ has degree less than $d(x)$.

Synthetic Division

• Efficient method for dividing by $x-c$.

Remainder Theorem

• If $f(x)$ is divided by $(x-c)$, the remainder is $f(c)$.

Factor Theorem

• $(x-c)$ is a factor of $f(x)$ if and only if $f(c) = 0$.

Zeros of Polynomials

Rational Zero Theorem

• If $\frac{p}{q}$ is a rational zero, $p$ divides the constant term and $q$ divides the leading coefficient.

Fundamental Theorem of Algebra

• A degree $n$ polynomial has exactly $n$ complex zeros (counting multiplicities).

Descartes’s Rule of Signs

• The number of positive real zeros equals the number of sign changes in $f(x)$ (or less by an even number).

• The number of negative real zeros equals the number of sign changes in $f(-x)$ (or less by an even number).