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Math1050 Ch05
Polynomial and Rational Functions
| Term | Definition |
|---|---|
| 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 |
Created by:
drjolley
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