Chapter 3

Vertex of a Parabola

(-b/2a, f(-b/2a))

Solve

1. Collect all terms to one side 2. Find the real solutions of the related equation f(x) = 0 and the values of x where f(x) is undefined. These are the "boundary" points for the solution set to the inequality. 3. Determine the sign of f(x) on the intervals defined by the boundary points. • If f(x) is positive, then the values of x on the interval are solutions to f(x) > 0. • If f (x) is negative, then the values of x on the interval are solutions to f(x) < 0. 4. Determine whether the boundary points are included in the solution set. 5. Write the solution set.

How to find Slant Asymptote

A rational function will have a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator. To find an equation of a slant asymptote of a rational function, divide the numerator by the denominator. The quotient will be linear and the slant asymptote will be of the form y = quotient.

How to determine if a given polynomial is a factor of f(x)

Apply the factor theorem and synthetic division

A quadratic function can be written in vertex form by?

Complete the square of a quadratic: f(x) = a(x - h)^2 + k

Conjugate Zeroes Theorem

Conjugate zeros theorem: If f(x) is a polynomial with real coefficients and if a + bi is a zero of f(x), then its conjugate a - bi is also a zero of f(x).

Vertex Formula to find Vertex of a Parabola

For f(x) = ax^2 + bx + c (a does not = 0), the vertex is given by (-b/2a, f(-b/2a))

Remainder Theorem

If a polynomial f(x) is divided by x - c, then the remainder is f(c). Note: The remainder theorem tells us that the value of f(c) is the same as the remainder we get from dividing f(x) by x - c.

Multiplicity

If a polynomial function f has a factor (x - c) that appears exactly k times, then c is a zero of multiplicity k. • If c is a real zero of odd multiplicity, then the graph of y = f(x) crosses the x-axis at c. • If c is a real zero of even multiplicity, then the graph of y = f(x) touches the x-axis (but does not cross) at c

Rational Zero Theorem

If f(x) = is a polynomial with integer coefficients and the coefficient on the leading term does not equal zero and p/q (in lowest terms) is a rational zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n greater than or equal to 1 with complex coefficients, then f(x) has at least one complex zero

Use of Remainder Theorem

If x-c divides a function with no remainder (remainder 0) then c is a zero of the function

Factor Theorem

Let f (x) be a polynomial. 1. If f(c) = 0, then (x - c) is a factor of f(x). 2. If (x - c) is a factor of f(x), then f(c) = 0.

Intermediate Value Theorem

Let f be a polynomial function. For a < b, if f(a) and f(b) have opposite signs, then f has at least one zero on the interval [a, b]. i.e. The function must cross the x-axis if the value of y changes from positive to negative or vice versa and the change must happen somewhere between the two values with opposite signs.

How to identify the horizontal asymptotes of a graph

Let n be the degree of the numerator and m be the degree of the denominator. 1. If n > m, then f has no horizontal asymptote. 2. If n < m, then the line y = 0 (the x-axis) is the horizontal asymptote of f. 3. If n = m, then the line y = a/b (where a is the lead coefficient of the numerator and b is the lead coeficient of the denominator) is the horizontal asymptote of f(x). Note: that n = m + 1 is a special case which results in a slant asymptote, not a vertical one

Division of Polynomials

Long division can be used to divide two polynomials Synthetic division is generally quicker/easier but can only be used to divide polynomials if the divisor is of the form x - c

How many vertical asymptotes might a rational function have

None, one, or many

If no headway can be made in factoring a polynomial what might you try?

See if rational zero theorem applies and if it does, list all possible rational zeroes (all combinations of factors of p over factors of q) One or more of these will be a zero of the function, you can then use a found zero to divide and factor the polynomial with the factor theorem

Find where/if the graph crosses its horizontal asymptote

Set f(x) = to the value of the horizontal asymptote and evaluate If you produce a contradiction, no crossing point exists because for no values of x does the function = the horizontal asymptote If you are able to solve for x then the crossing point is (x, horizontal asymptote)

How to determine End Behavior of a polynomial function?

The far left and far-right behavior of the graph of a polynomial function is determined by the leading term of the polynomial The degree of the polynomial and the sign of the leading coefficient

Quadratic Function

The function defined by f(x) = ax^2 + bx + c (a does not equal 0)

What happens if a rational function has common factors in its numerator and denominator?

The function produces a hole, 0/0 where the value of y becomes undefined.

Maximum Turning Points

The graph of a polynomial function of degree n will have at most n - 1 turning points

Key elements of Quadratic Function

The graph of a quadratic function forms a parabola f(x) = a(x - h)^2 + k • The vertex is (h, k). • If a > 0, the parabola opens upward, and the minimum value of the function is k. • If a < 0, the parabola opens downward, and the maximum value of the function is k. • The axis of symmetry is the line x = h. • The x-intercepts are determined by the real solutions to the equation f(x) = 0. • The y-intercept is determined by f(0).

Real Solutions to f(x)

The zeros of a polynomial function defined by y = f(x) are the values of x in the domain of f for which f(x) = 0. These are the real solutions (or roots) of the equation f(x) = 0.

Build a polynomial with degree 3 and known zeroes at 1, 6, and -3

Use the factor theorem, for each zero the polynomial has a factor (x - (c)) (x - 1)(x - 6)(x - (-3)) = x^3 - 2x^2 - 15x + 18

f(x) = a(x - h)^2 + k

Vertex form of a Quadratic Function

How do we find the vertical asymptotes of a rational function?

With the rational function in lowest terms Find all values for which the x = 0 in the denominator while the numerator is non zero. This is easiest to do by setting the denominator = 0 and factoring it in order to use the zero product property. Make sure to double-check that the function is in lowest terms

Steps to Graph a Quadratic Function

a. Determine whether the graph of the parabola opens upward or downward. b. Identify the vertex. c. Determine the x-intercept(s). d. Determine the y-intercept. e. Sketch the function. f. Determine the axis of symmetry. g. Determine the maximum or minimum value of f. h. Write the domain and range in interval notation.

If c is a zero of odd multiplicity, then the graph...?

crosses the x-axis at c. The point (c, 0) is called a cross point.

End Behavior if: Degree is even, and Leading Coefficient is positive

f(x) approaches infinity as x approaches + or - infinity

End Behavior if: Degree is odd, and Leading Coefficient is positive

f(x) approaches infinity as x approaches infinity, f(x) approaches negative infinity as x approaches negative infinity

End Behavior if: Degree is even, and Leading Coefficient is negative

f(x) approaches negative infinity as x approaches + or - infinity

End Behavior if: Degree is odd, and Leading Coefficient is negative

f(x) approaches negative infinity as x approaches infinity, f(x) approaches infinity as x approaches negative infinity

Factor a polynomial given that -5 is a known zero

if f(-5) = 0 then (x - (-5)) is a factor of f(x) Using synthetic division, -5 divides the polynomial resulting in some quotient q(x) which is a factor of f(x) f(x) = (x - (-5))(q(x)) This newly factored form of f(x) can then be used to solve for additional zeroes by other means

If c is a zero of even multiplicity, then the graph...?

touches the x-axis at c and turns back around (does not cross the x-axis). The point (c, 0) is called a touch point.