Ch 5 Overview: Polynomial and Rationa Functions

What is another use of the remainder theorem

to test whether a rational number is a zero for a given polynomial.

continuous function

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper.

axis of symmetry

A line that divides a plane figure or a graph into two congruent reflected halves; A parabola is symmetric about the vertex

power function

A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number.

smooth curve

A smooth curve is a graph that has no sharp corners.

Define turning point.

A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

turning point

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.

Factor Theorem

According to the Factor Theorem, k is a zero of f(x) if and only if (x−k) is a factor of f(x)

What is the domain of quadratic functions?

All real numbers unless the function is restricted.

Division Algorithm

Is a method of looking at a solution to a division problem: dividend = (divisor⋅quotient) + remainder

Define the intermediate value theorem.

Let f be a polynomial function. The Intermediate Value Theorem states that if f(a) and f(b) have opposite signs, then there exists at least one value c between a and b for which f(c)=0.

Why does the rational zeros theorem work?

Lets consider a function with two zeros as follows: x=2/5 and x=3/4 Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4. We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.

Is absolute value function a polynomial?

No, because it has a sharp curve.

Is f(x) = 2^x a power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.

The coefficient of the leading coefficient must be what to divide polynomials using synthetic division?

ONE

What are the components of a power function?

1. A real number 2. A Coefficient 3. A variable raised to a fixed real number

List the steps that occur , given a polynomial and a binomial, use long division to divide the polynomial by the binomial.

1. Set up the division problem 2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. 3. Multiply the answer by the divisor and write it below the like terms of the dividend. 4.Subtract the bottom binomial from the top binomial. 5. Bring down the next term of the dividend. 6. Repeat steps 2-5 until reaching the last term of the dividend. 7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.

What is one important feature of a quadratic function?

One important feature of the graph is that it has an extreme point, called the vertex

What is the largest amount of turning points a function will have based on its leading term

. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

From the standard form of a quadratic function, what do the values h and k represent?

(h, k) is the vertex.

The vertex is :

(h,K)

How do we determine the highest degree and leading coefficient?

1. Find the highest power of x to determine the degree function. 2. Identify the term containing the highest power of x to find the leading term. 3. Identify the coefficient of the leading term.

List the steps of graphing polynomials.

1. Find x & y intercepts 2. Check for symmetry. 3. Use the multiplicities of zero to determine end behavior at the x-intercepts 4. Determine end behavior using leading term 5. Use the end behavior and the behavior at the intercepts to sketch a graph. 6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial.

How do you, given a graph of a polynomial function, write a formula for the function.

1. Identify the x-intercepts of the graph to find the factors of the polynomial. 2. Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor. 3. Find the polynomial of least degree containing all the factors found in the previous step. 4. Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.

Name three GRAPHICAL BEHAVIORs OF POLYNOMIALS AT X-INTERCEPTS

1. If a polynomial contains a factor of the form (x−h)p, the behavior near the x-intercept h is determined by the power p. We say that x=h is a zero of multiplicity p. 2. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities. 3, The sum of the multiplicities is the degree of the polynomial function.

Given a polynomial function f, how do you use synthetic division to find its zeros.

1. Use the Rational Zero Theorem to list all possible rational zeros of the function. 2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate. 3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic. 4. Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.

HOW TO Given the zeros of a polynomial function f and a point (f(c) on the graph off, f, use the Linear Factorization Theorem to find the polynomial function.

1. Use the zeros to construct the linear factors of the polynomial. 2. Multiply the linear factors to expand the polynomial. 3.Substitute (c,f(c)) into the function to determine the leading coefficient. 4. Simplify.

Given two polynomials list the steps to use synthetic division to divide.

1. Write k for the divisor. 2. Write the coefficients of the dividend. 3. Bring the lead coefficient down. 4. Multiply the lead coefficient by k. Write the product in the next column. 4. Add the terms of the second column. 6. Multiply the result k. Write the product in the next column. 7.Repeat steps 5 and 6 for the remaining columns. 8.Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.

What is the range of a quadratic function? Why?

Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.

Why do we look at the leading term with the highest exponent to determine end behavior?

Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term

What device mimics a parabola shape?

Curved antennas, such as the ones shown in Figure 1, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

When does a rational function not have any horizontal asymptotes but a slant asymptote?

Degree of numerator > Degree of denominator by one Numerator > Denominator Wh

If a <0 the parabola opens

Downward

What does each turning point of a polynomial function represent?

Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph

What does knowing the degree of a polynomial help us predict?

End Behavior

Sate what the graph of a function does at the following: 1. Even multiplicity 2. Odd Multiplicity

Even: Touch x- axis Odd: cross the x- axis

THE REMAINDER THEOREM

If a polynomial f(x) is divided by x-k, then the remainder is the value f(k)

If the parabola opens up, the vertex represents the ____________ __________ on the graph, or the ___________ value of the quadratic function. If the parabola opens down, the vertex represents the _____________ ___________ on the graph, or the ____________ value.

If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.

x- intercepts represent ....

If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y=0.

The vertex is considered a ____________ __________ of the graph

In either case, the vertex is a turning point on the graph

Define positive and negative infinity utilizing symbol notation

Positive Infinity: +∞ Negative Infinity: −∞

Define synthetic division.

Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.

What method other than long- hand division can we divide polynomials by?

Synthetic division.

What is the sum of the multiplicities

The degree of a polynomial

Other than end behavior, what else can the degree of a polynomial help us determine?

The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points.

degree

The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.

How many turning points will a graph possess with a degree of n.

The graph of the polynomial function of degree n must have at most n-1 turning points. Think; the polynomial will have at most the same number of zeros (x- intercepts) where y=0 The x-intercepts determine the turning points of a graph. When we graph the x-intercepts, we see we can only have n-1 turning points

As the power becomes larger in value, what does the graph do?

The graphs flatten somewhat near the origin and become steeper away from the origin.

Vertex of a parabola

The highest or lowest point on the parabola

leading term

The leading term is the term containing the highest power of the variable, or the term with the highest degree.

Define multiplicity

The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity.

Even power functions all possess..

The same graph

What is the standard form of a quadratic equation useful for?

The standard form is useful for determining how the graph is transformed from the graph of y=x²

What occurs to the y-value at a turning point? The X - values?

The why value is the point at which the function has an input value of zero. The x- intercepts are the points at which the output value is zero.

x intercept of a parabola

The x-intercepts are the points at which the parabola crosses the x-axis.

y-intercept of a parabola

The y-intercept is the point at which the parabola crosses the y-axis.

What does the following read as ? x→∞

This is read as "x approaches infinity"

How do we determine end behavior of a polynomial?

To determine its end behavior, look at the leading term of the polynomial function.

What is the form of a power function?

f(x)= kxⁿ where k and n are real numbers, and k is known as the coefficient.

The standard form of a quadratic equation is:

f(x)=a(x−h)²+k

A quadratic function is a polynomial function of degree _______

Two

What is the shape of a quadratic function?

U-Shaped

If a > 0 then the parabola opens

Upward

How do we mitigate dividing by a number, such as 2 (as we would in division of whole numbers) then multiplying and subtracting the middle product

We change the sign of the divisor

Once we utilize the rational zeros theorem to narrow down all of the possible zeros of the function, what do we do?

We use synthetic division to determine if the rational numbers are zero of the function. We know this, if there is no remainder when dividing.

Using the remainder theorem to test rational numbers sounds awesome, but how do we know which rational number to test for zeros of the polynomial?

We use the Rational Zero Theorem

Other than end behavior, what do we want to know about the graph?

What happens in the middle of the function; In particular, we are interested in locations where graph behavior changes.

Aside from defining end behavior, we can also determine a polynomial functions local behavior. What can we tell about local behavior from multiplicity?

Whether the graph touches and continues in the same direction or crosses the x axis changing direction

Do all polynomial functions have as their domain all real numbers?

Yes. Any real number is a valid input for a polynomial function.

standard form of a quadratic function

f(x)=a(x−h)²+k

Finish the statement: The graphs of polynomial functions are both...

continuous and smooth.

The general form of a quadratic function

f(x) = ax² + bx + c

general form of a quadratic function

f(x) = ax² + bx + c

The fundamental theorem of algebra is the basis for doing what?

forms the foundation for solving polynomial equations

Finish the statement: A polynomial function of nth degree is the product of n factors, so it will have_____________________.

have at most n roots or zeros, or x-intercepts.

How many terms is a power function?

one term

The graph of a quadratic function is a

parabola

What do quadratic functions frequently model?

problems involving area and projectile motion

Finish the statement: Turning points of a smooth graph must be...

smooth graph must always occur at rounded curves

What does the fundamental theorem of algebra tell us?

that every polynomial function has at least one complex zero.

What is a zero or root in mathematics?

the values of x at which y=0.

What can we confirm from the following: In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, what can we confirm? Why?

there is a zero between them

What does infinity mean when referencing end behavior ?

we are saying that x is increasing without bound.

How do we describe end behavior as numbers become larger and larger?

we use the idea of infinity.

What is the axis of symmetry defined as?

x= -b/2a

The axis of symmetry is given by:

x=-b/2a