Polynomial and Rational Functions
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
|
|
||||
|---|---|---|---|---|---|
| T or F? Polynomial functions must have powers of nonnegative integers. | T
🗑
|
||||
| Graphs of polynomial functions must be both ___ and ___. | continuous, smooth
🗑
|
||||
| If f(r)=0 then r is a ___. | real zero
🗑
|
||||
| Real zeros are ___-intercepts. | x
🗑
|
||||
| For real zeros r, ___ is a factor of f. | x-r
🗑
|
||||
| For real zeros r, r is a solution to f(x)=___. | 0
🗑
|
||||
| Power functions are a type of ___ function. | polynomial
🗑
|
||||
| Power functions are in the form f(x)=___. | ax^n
🗑
|
||||
| f(x)=3x is a power function of degree ___. It is a ___. Its y-intercept is ___. | 1, straight line, 0
🗑
|
||||
| f(x)=-5x^2 is a power function of degree ___. It is a ___. Its y-intercept is ___. Vertex is at ___. It opens ___. | 2, parabola, 0, (0,0), down
🗑
|
||||
| For power functions of even degree, the domain is ___ and the range is ___. | all real numbers, all nonnegative numbers
🗑
|
||||
| For power functions of even degree, the graph always contains points ___, ___, ___. | origin, (-1,1), (1,1)
🗑
|
||||
| For power functions of odd degree, the graph is symmetric to the ___. | origin
🗑
|
||||
| For power functions of even degree, the graph is symmetric to the ___. | y-axis
🗑
|
||||
| For power functions of odd degree, the domain is ___ and the range is ___. | all real numbers, all real numbers
🗑
|
||||
| For power functions of odd degree, the graph always contains points ___, ___, ___. | origin, (-1,-1), (1,1)
🗑
|
||||
| For zeros of even multiplicity, the graph ___ the x-axis. | touches
🗑
|
||||
| For zeros of odd multiplicity, the graph ___ the x-axis. | crosses (hint: cross-dressers are odd)
🗑
|
||||
| If f is a polynomial function of degree 3, the graph has AT MOST ___ turning points. | 2, (could be fewer though), i.e., n-1 TPs
🗑
|
||||
| For large values of |x|, f(x)=ax^5+bx^4+... resembles the graph ___. This is called its ___. | y=ax^5, end behavior
🗑
|
||||
| For large values of |x|, f(x)=x^2(x-2) resembles the graph ___. | y=x^3
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 1 | Determine the end behavior, rewrite in standard form if necessary
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 2 | Find the x & y intercepts
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 3 | determine the zeros and their multiplicity and determine whether they cross or touch the x-axis
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 4 | determine the max TPs
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 5 | determine the behavior near each zero
🗑
|
||||
| Analyzing the graph of a polynomial function: Step 6 | draw
🗑
|
||||
| Ratios of polynomials are called ___. | rational functions
🗑
|
||||
| Rational function are said to be in lowest terms if they have no ___. | common factors
🗑
|
||||
| T or F? Asymptotes are always linear functions. | T
🗑
|
||||
| ___ asymptotes are the reals zeros of the denominator. | Vertical
🗑
|
||||
| To find asymptotes, the function must first be in ___. | lowest terms
🗑
|
||||
| In lowest terms, a rational fxn is said to be ___ if the degree of the numerator is less than that of the denominator. | proper
🗑
|
||||
| All proper functions have the horizontal asymptote ___. | y=0
🗑
|
||||
| In lowest terms, a rational fxn is said to be ___ if the degree of the numerator equal to or more than that of the denominator. | improper
🗑
|
||||
| To find horizontal and oblique asymptotes of improper fxns, we use long division to write the rational fxn as the ___ of the polynomial (the quotient) plus a proper rational fxn (the remainder over the denominator). | sum,
🗑
|
||||
| If the polynomial quotient of a rational fxn is f(x)=b, a constant, then the line ___ is a ___ asymptote. | y=b, horizontal
🗑
|
||||
| If the polynomial quotient of a rational fxn is f(x)=ax+b, then the line ___ is a ___ asymptote. | y=ax+b, oblique
🗑
|
||||
| In lowest terms, if the numerator and denominator are of the same degree in a rational fxn, the quotient of the ___ will give us the ___ asymptote. | leading coefficients, horizontal
🗑
|
||||
| In lowest terms, if the quotient of a rational function is not ___, it has neither a horizontal nor oblique asymptote. | linear
🗑
|
||||
| Steps for finding horizontal & oblique asymptotes: The rational fxn must first be ___. | in lowest terms
🗑
|
||||
| Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n<m, R is ___ and it has a/an ___ asymptote which is ___. | proper, horizontal, y=0
🗑
|
||||
| Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n=m, R is ___ and it has a/an ___ asymptote which is ___. | improper, horizontal, y=a/b (the ratio of the leading coefficients)
🗑
|
||||
| Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n=m+1, R is ___ and it has a/an ___ asymptote which is ___. | improper, oblique, in the form y=ax+b
🗑
|
||||
| Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n = m + 2 or more, R is ___ and it has ___ asymptote and the graph behaves like the graph of the ___ for large values of |x|. | improper, no horizontal or oblique, quotient
🗑
|
||||
| The graph of a rational fxn has either ___ horizontal or ___ oblique asymptote or it has ___. | 1, 1, neither
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 1 | Factor the fxn and find the domain of the denominator.
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 2 | Write R in lowest terms
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 3 | Locate the intercepts of R, determine the end behavior at each x-intercept
🗑
|
||||
| The x-intercepts of rational fxn R are the zeros of the ___ that are in the domain of R. | numerator
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 4 | Determine and graph the vertical asymptotes
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 5 | Determine and graph the horizontal or oblique asymptote (if there is one). Determine if the graph intersects the asymptote. Plot these points.
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 6 | Use the zeros to divide the x-axis into intervals. Determine where the graph is above and below by plotting points.
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 7 | Analyze and graph the behavior near each asymptote.
🗑
|
||||
| Analyzing the graph of a rational fxn R: Step 8 | Graph
🗑
|
||||
| Solving a polynomial inequality: Step 1 | Write the inequality with a zero on one side. i.e., f(x)<0, f(x)>0, f(x)< or = 0, f(x)> or = 0
🗑
|
||||
| Solving a polynomial inequality: Step 2 | Determine the zeros and if it is rational, where the fxn is undefined.
🗑
|
||||
| Solving a polynomial inequality: Step 3 | Separate x into intervals.
🗑
|
||||
| Solving a polynomial inequality: Step 4 | Determine where f(x) is + or - for each interval. BE CAREFUL TO EXCLUDE WHERE f IS UNDEFINED.
🗑
|
||||
| If f(r)=0 then ___ is an x-intercept, ___ is a factor of f, and r is a solution to the equation ___. | r, x-r, f(x)=0
🗑
|
||||
| The remainder theorem says the if f(x) is divided by x-c, then the remainder is ___. | f(c)
🗑
|
||||
| The factor theorem says that: 1. If f(c)=0 then ___ is a factor of ___. 2. If ___ is a factor of f(x), then ___. | x-c, f(x), x-c, f(c)=0
🗑
|
||||
| A polynomial function cannot have more real zeros than ___. | its degree
🗑
|
||||
| Rational zeros theorem: For a polynomial fxn R in standard form, if p/q is a factor of R then p must be a factor of ___ and q must be a factor of ___. | The last coefficient, the leading coefficient
🗑
|
||||
| T or F? At least one of the possible zeros from the list of the possible rational zeros must be a zero. | T
🗑
|
||||
| The quotient of a fxn R and a factor is called a ___ equation. | depressed
🗑
|
||||
| f(x)= x^3 + 3x^2 + 4x + 12 can have at most ___ zeros. | 3
🗑
|
||||
| For f(x)= 2x^3 + 3x^2 + 4x + 12, possible zeros are ___. | + or - (1, 2, 3, 4, 6, 12, 1/2, 3/2)
🗑
|
||||
| Intermediate value theorem says that if a<b and f(a) and f(b) are of ___ sign, there is at least one real zero between a and b. | opposite
🗑
|
||||
| T or F? Complex polynomials have the same number of linear factors as the degree. | T
🗑
|
||||
| For complex polynomials, if r=a+bi is a zero then ___ is also a zero. This is called its ___. | a-bi, complex conjugate
🗑
|
||||
| (a-bi)(a+bi)=___ | a^2 + b^2
🗑
|
||||
| A polynomial of ___ degree with real coefficients has at least one real zero. | odd
🗑
|
Review the information in the table. When you are ready to quiz yourself you can hide individual columns or the entire table. Then you can click on the empty cells to reveal the answer. Try to recall what will be displayed before clicking the empty cell.
To hide a column, click on the column name.
To hide the entire table, click on the "Hide All" button.
You may also shuffle the rows of the table by clicking on the "Shuffle" button.
Or sort by any of the columns using the down arrow next to any column heading.
If you know all the data on any row, you can temporarily remove it by tapping the trash can to the right of the row.
To hide a column, click on the column name.
To hide the entire table, click on the "Hide All" button.
You may also shuffle the rows of the table by clicking on the "Shuffle" button.
Or sort by any of the columns using the down arrow next to any column heading.
If you know all the data on any row, you can temporarily remove it by tapping the trash can to the right of the row.
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.
Normal Size Small Size show me how
Normal Size Small Size show me how
Created by:
drjolley
Popular Math sets