1. A quadratic function f is a function of the form

   $$f(x)=a{x}^2+bx+c,a\ne 0.$$

2. The standard form of a quadratic function is

   $$f(x)=a{(x-h)}^2+k,a\ne 0.$$

3. The graph of a quadratic function is a transformation of the graph of $y=x^2.$

4. The graph of a quadratic function is a parabola with vertex

   $$(h,k)=(-{\displaystyle \frac{b}{2a}},f(-{\displaystyle \frac{b}{2a}}\left)\right).$$

5. The maximum (if $a<0$) or minimum (if $a>0$) value of a quadratic function $f(x)=a{x}^2+bx+c$ is the y-coordinate, k, of the vertex of the parabola.

1. A function f of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots a_2{x}^2+a_{1}x+a_0,$ $a_n\ne 0,$ is a polynomial function of degree n.

2. The graph of a polynomial is smooth and continuous.

3. The end behavior of the graph of a polynomial function depends on the sign of the leading coefficient and the degree (even-odd) of the polynomial.

4. A real number c is a zero of a function f if $f(c)=0.$ Geometrically, c is an x-intercept of the graph of $y=f(x).$

5. If in the factorization of a polynomial function f the factor $(x-a)$ occurs exactly m times, then a is a zero of multiplicity m. If m is odd, the graph of $y=f(x)$ crosses the x-axis at a; if m is even, the graph touches but does not cross the x-axis at a.

6. If the degree of a polynomial function f is n, then f has, at most, n real zeros and the graph of f has, at most, $n-1$ turning points.

7. Intermediate Value Theorem. Let f be a polynomial function and a and b be two numbers such that $a<b.$ If f(a) and f(b) have opposite signs, then there is at least one number c, with $a<c<b,$ for which $f(c)=0.$

8. See Example 9 on page 342 for the procedure for graphing a polynomial function.

1. Division Algorithm. If a polynomial F(x) is divided by a polynomial $D(x)\ne 0,$ there are unique polynomials Q(x) and R(x) such that $F(x)=D(x)Q(x)+R(x),$ where either $R(x)=0$ or deg $R(x)<\text{deg D}(x).$ In words, “The dividend equals the product of the divisor and the quotient plus the remainder.”

2. Synthetic division is a shortcut for dividing a polynomial $F(x)\text{ by }x-a.$

3. Remainder Theorem. If a polynomial F(x) is divided by $x-a,$ the remainder is F(a).

4. Factor Theorem. A polynomial function F(x) has $x-a$ as a factor if and only if $F(a)=0.$

1. Rational Zeros Theorem. If ${\displaystyle \frac{p}{q}}$ is a rational zero in lowest terms for a polynomial function with integer coefficients, then p is a factor of the constant term and q is a factor of the leading coefficient.

2. Descartes’s Rule of Signs. Let F be a polynomial function with real coefficients.

   1. The number of positive zeros of F is equal to the number of variations of signs of F(x) or is less than that number by an even integer.

   2. The number of negative zeros of F is equal to the number of variations of sign of $F(-x)$ or is less than that number by an even integer.

3. Rules for Bounds on the Zeros. Suppose a polynomial F is synthetically divided by $x-k.$

   1. If $k>0$ and each number in the last row is zero or positive, then k is an upper bound on the zeros of F.

   2. If $k<0$ and the numbers in the last row alternate in sign, then k is a lower bound on the zeros of F.

1. Fundamental Theorem of Algebra. An nth-degree polynomial equation has at least one complex zero.

2. Factorization Theorem for Polynomials. If P(x) is a polynomial of degree $n\ge 1,$ it can be factored into n (not necessarily distinct) linear factors in the form $P(x)=a(x-r_1){(x-r_2)}_{}\dots (x-r_n),$ where $a,r_1,r_2,\dots ,r_n$ are complex numbers.

3. Number of Zeros Theorem. A polynomial of degree n has exactly n complex zeros, provided a zero of multiplicity k is counted k times.

4. Conjugate Pairs Theorem. If $a+bi$ is a zero of the polynomial function P (with real coefficients), then $a-bi$ is also a zero of P.

1. A function $f(x)={\displaystyle \frac{N(x)}{D(x)}}$, where N(x) and D(x) are polynomials and $D(x)\ne 0,$ is called a rational function. The domain of f is the set of all real numbers except the real zeros of D(x).

2. The line $x=a$ is a vertical asymptote of the graph of f if $|f(x)|\to \infty$ as $x\to a^{+}$ or as $x\to a^{-}$.

3. If ${\displaystyle \frac{N(x)}{D(x)}}$ is in lowest terms, then the graph of f has vertical asymptotes at the real zeros of D(x).

4. The line $y=k$ is a horizontal asymptote of the graph of f if $f(x)\to k$ as $x\to \infty$ or as $x\to -\infty$.

5. The line $y=mx+b$ is an oblique asymptote of the graph of f if the degree of N(x) is exactly one more than the degree of D(x).

6. A procedure for graphing rational functions is given on page 392.

k is a nonzero constant called the constant of variation.