| 单词 | asymptote |
| 例句 | Hynes’s genius is the way he keeps the anxiety fueled by 9/11 in the margins until the story finally hurtles us down the asymptote of terror. Perspective | How 9/11 altered the fiction landscape in 13 novels 2021-09-05T04:00:00Z “One line is curved, the other line is straight,” Ms. Kidwell explained, “but the principle of the asymptote is that they’re never going to meet.” Playing a ‘Game’ to Reveal Uncomfortable Truths About Race 2016-09-07T04:00:00Z “After three dates, he still seems afraid of physical contact. I finally told him: “If you’re just going to approach me but never touch me, you can take your *asymptote it home.” Style Invitational Week 1172: Pieces of ‘Pie’ 2016-04-21T04:00:00Z The asymptote of his music is a punctuated silence, which is why he was especially sensitive to his drummers and dependent on them to organize the music’s forward motion. The Best of Thelonious Monk 2015-05-20T04:00:00Z To judge by their opposing interpretations of a metaphor borrowed from geometry — the asymptote, whose lines approach without ever intersecting — the answer to that might not be an easy yes. Playing a ‘Game’ to Reveal Uncomfortable Truths About Race 2016-09-07T04:00:00Z Still, Shepard dares to move his narrative down the asymptote of despair, and the moral heroism he describes on that path toward infinity, you will never forget. ‘The Book of Aron,’ by Jim Shepard, is a masterpiece 2015-04-28T04:00:00Z The asymptote is a line that a curve approaches as it stretches along a graph’s axes. One Year of #MeToo: We Need a More Inclusive Language of Abuse and Victimhood 2018-10-10T04:00:00Z But there is one concept that I think I understand: the asymptote. One Year of #MeToo: We Need a More Inclusive Language of Abuse and Victimhood 2018-10-10T04:00:00Z The curve tends toward positive or negative infinity, always approaching the asymptote but never reaching it. One Year of #MeToo: We Need a More Inclusive Language of Abuse and Victimhood 2018-10-10T04:00:00Z It’s clear he now believes this arc, at best, reaches an asymptote — that dastardly dotted line it can never quite touch. Through the Lens of the Obama Years, Ta-Nehisi Coates Reckons With Race, Identity and Trump 2017-10-01T04:00:00Z Do they all have the same vertical asymptote? College Algebra with Corequisite Support 2020-09-23T00:00:00Z The function and the asymptotes are shifted 3 units right and 4 units down. College Algebra 2021-12-21T00:00:00Z Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. Intermediate Algebra 2020-05-06T00:00:00Z Find the horizontal asymptote and interpret it in context of the problem. Precalculus 2021-12-01T00:00:00Z Sketch asymptotes as a guide to the graph. Calculus, Volume 3 2016-03-30T00:00:00Z What is the equation of the vertical asymptote? College Algebra with Corequisite Support 2020-09-23T00:00:00Z An asymptote is a line that the graph of a function approaches, as either increases or decreases without bound. College Algebra 2021-12-21T00:00:00Z Sketch the asymptotes—the lines through the diagonals of the rectangle. Intermediate Algebra 2020-05-06T00:00:00Z Notice the horizontal asymptote is This means the concentration, the ratio of pounds of sugar to gallons of water, will approach 0.1 in the long term. Precalculus 2021-12-01T00:00:00Z The asymptotes of this hyperbola are the x and y coordinate axes. Calculus, Vol 2 2016-03-30T00:00:00Z The line y = 0 is a horizontal asymptote for y = 2 x. Calculus, Volume 1 2016-03-30T00:00:00Z The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small. College Algebra 2021-12-21T00:00:00Z Sketch the asymptotes—the lines through the diagonals of the rectangle. Intermediate Algebra 2020-05-06T00:00:00Z This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. Algebra and Trigonometry 2015-02-13T00:00:00Z The red dashed line represents the carrying capacity, and is a horizontal asymptote for the solution to the logistic equation. Calculus, Vol 2 2016-03-30T00:00:00Z The line x = a is called a vertical asymptote of the graph. Calculus, Volume 1 2016-03-30T00:00:00Z No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers. College Algebra 2021-12-21T00:00:00Z Notice that the horizontal asymptote also shifted down 2 units. Intermediate Algebra 2020-05-06T00:00:00Z Identify the horizontal and vertical asymptotes of the graph, if any. Algebra and Trigonometry 2015-02-13T00:00:00Z Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote. Calculus, Vol 2 2016-03-30T00:00:00Z We formally define a vertical asymptote as follows: Calculus, Volume 1 2016-03-30T00:00:00Z The important question is the value of the effort, and whether it leaves us separated by an asymptote or a chasm. Opinion | The Trouble With Empathy 2020-09-04T04:00:00Z When the graph approaches the y-axis so very closely but will never cross it, we call the line x = 0, the y-axis, a vertical asymptote. Intermediate Algebra 2020-05-06T00:00:00Z TRY IT #2 Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units. Algebra and Trigonometry 2015-02-13T00:00:00Z It’s like an asymptote, moving toward but never arriving at the point of convergence. Affect Theory and the New Age of Anxiety 2019-03-18T04:00:00Z In other words, the function has a vertical asymptote at v/c = 1. Calculus, Volume 1 2016-03-30T00:00:00Z As more people with autism receive genetic diagnoses, however, this growth spurt may approach its asymptote and, ultimately, clear a path for the use of DBS in some instances. Can brain-stimulating implants treat some severe cases of autism? 2018-09-14T04:00:00Z The a, b values in the equation also help us find the asymptotes of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z The events at VW manifest, rather melodramatically, this approaching asymptote, where there is no more blood to be squeezed from the stone. How Tesla Leaves its Rivals Playing Catch Up 2015-10-08T04:00:00Z Back in Introduction to Functions and Graphs, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z As still to its asymptote speedeth the curve, So approximates Man—Thee, who reachable not, Hast formed him to yearningly Follow thy whole Sole and single omniscience!” Browning and His Century 2012-02-15T03:00:39.033Z Sketch the asymptotes—the lines through the diagonals of the rectangle. Intermediate Algebra 2020-05-06T00:00:00Z However, the graph of looks like a diagonal line, and since will behave similarly to it will approach a line close to This line is a slant asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Figure 4.40 The function approaches the asymptote y = 2 as x approaches ±∞. Calculus, Volume 1 2016-03-30T00:00:00Z "Divine perfection is the asymptote of human life, toward which it constantly strives, to which it draws nearer and nearer, but which can be attained only at infinity." Anarchism 2011-07-12T02:00:35.053Z Sketch the asymptotes—the lines through the diagonals of the rectangle. Intermediate Algebra 2020-05-06T00:00:00Z To find the equation of the slant asymptote, divide The quotient is and the remainder is 2. Algebra and Trigonometry 2015-02-13T00:00:00Z For the asymptotes as axes of co-ordinates the equation of the hyperbola is xy = const. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z However, a function may cross a horizontal asymptote. Calculus, Volume 1 2016-03-30T00:00:00Z The area of the loop, which equals the area between the curve and its asymptote, is 3a/2. Encyclopaedia Britannica, 11th Edition, Volume 10, Slice 5 "Fleury, Claude" to "Foraker" 2011-04-03T02:00:20.883Z Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes. Intermediate Algebra 2020-05-06T00:00:00Z The slant asymptote is the graph of the line See Figure 13. Algebra and Trigonometry 2015-02-13T00:00:00Z If the chord is changed into a tangent, this gives— The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z In fact, a function may cross a horizontal asymptote an unlimited number of times. Calculus, Volume 1 2016-03-30T00:00:00Z But after you have demonstrated to him the properties of the hyperbola and its asymptote, the apparent absurdity vanishes. The Religion of Geology and Its Connected Sciences 2011-02-28T03:00:28.890Z The y -axis is again the vertical asymptote. Intermediate Algebra 2020-05-06T00:00:00Z As ∞ resulting in a horizontal asymptote at See Figure 14. Algebra and Trigonometry 2015-02-13T00:00:00Z Hence— Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Figure 4.53 The graph of this rational function approaches a horizontal asymptote as x → ±∞. Calculus, Volume 1 2016-03-30T00:00:00Z Your existence for that time being, was like an asymptote running along, intersecting and exceeding everything. Poetic Connections: When Love Is in Transit 2010-12-05T03:20:00Z Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions. Intermediate Algebra 2020-05-06T00:00:00Z Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the 701 hyperbola. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Figure 4.54 The graph of this rational function approaches the horizontal asymptote y = 0 as x → ±∞. Calculus, Volume 1 2016-03-30T00:00:00Z The vertex was, however, in reality rounded off, and the figure rather resembled a hyperbola, which tended to coincidence with its asymptotes. The Glaciers of the Alps Being a narrative of excursions and ascents, etc. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the x, y values get larger and larger. Intermediate Algebra 2020-05-06T00:00:00Z Then, find the x- and y-intercepts and the horizontal and vertical asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Figure 4.57 This function has two horizontal asymptotes and it crosses one of the asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z The wise man, an ancient writer has said, is to the woman what the asymptote is to the curve, it draws ever nearer but never touches. Underground Man The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Since f is a polynomial function, it does not have any vertical asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z The curve is symmetrical about the axis of y and has the axis of x for its asymptote. Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 7 "Columbus" to "Condottiere" The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z For the vertical asymptote at the factor was not squared, so the graph will have opposite behavior on either side of the asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z From this the following theorem is a simple deduction: All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z To determine whether f has any vertical asymptotes, first check to see whether the denominator has any zeroes. Calculus, Volume 1 2016-03-30T00:00:00Z The area between the curve and its asymptote is 3πa�, i.e. three times the area of the generating circle. Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 4 "Cincinnatus" to "Cleruchy" The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. Intermediate Algebra 2020-05-06T00:00:00Z The asymptote at is exhibiting a behavior similar to with the graph heading toward negative infinity on both sides of the asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z In particular a circle x� + y� = a�, a particular ellipse, has for asymptotes the imaginary lines x = �y √−1. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Since the degree of the numerator is one more than the degree of the denominator, f must have an oblique asymptote. Calculus, Volume 1 2016-03-30T00:00:00Z To-day is but the asymptote of to-morrow, that curve perpetually drawing near, but never reaching the straight line flying into infinity. The Atlantic Monthly, Volume 18, No. 107, September, 1866 Sketch the asymptotes through the diagonals of the rectangle. Intermediate Algebra 2020-05-06T00:00:00Z Can a graph of a rational function have no vertical asymptote? Algebra and Trigonometry 2015-02-13T00:00:00Z The hyperbola would have an equation of the form xy = c if referred to its asymptotes as axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The line y = x + 1 is an oblique asymptote for f . Calculus, Volume 1 2016-03-30T00:00:00Z Even so there are asymptote figures in geometry where an infinite length makes only a finite progress in breadth. Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil Therefore The asymptotes cut off on each tangent a segment which is bisected by the point of contact. An Elementary Course in Synthetic Projective Geometry For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Truth be told, I hated calculus, hated Isaac Newton and asymptotes and the whole smelly business. A Place so Foreign To check for vertical asymptotes, look at where the denominator is zero. Calculus, Volume 1 2016-03-30T00:00:00Z Then let the Eye of Science and her parabolas and hyperbolas, and conjugates, and asymptotes, and the rest of the confounded nonsensical farrago, all go to pot! All Around the Moon Given the asymptotes and a finite point of a conic, to construct the conic. An Elementary Course in Synthetic Projective Geometry For the following exercises, find the slant asymptote of the functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Risks on this score, however, will become less formidable with the further advance of practical astronomy along a track definable as an asymptote of ideal perfection. Scientific American Supplement, No. 829, November 21, 1891 Step 4: The function has no vertical asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z "Then produce your asymptote!" exclaimed Barbican, with an angry sneer. All Around the Moon As previously noted, these two tangents are called the asymptotes of the curve. An Elementary Course in Synthetic Projective Geometry For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Algebra and Trigonometry 2015-02-13T00:00:00Z P will therefore be infinitely remote, and the point I having then reached t, where D T touches the smaller circle, C t S will be an asymptote to the curve. Scientific American Supplement, No. 803, May 23, 1891 Then, calculate the actual horizontal asymptote or limit. Calculus, Volume 1 2016-03-30T00:00:00Z Any persons whom it could please have no better notion of what the words referred to signify than of the meaning of apsides and asymptotes. The Atlantic Monthly, Volume 03, No. 20, June, 1859 If the chord A"B" meet the asymptotes in A', B', then A', P', B', and the point at infinity are four harmonic points, and therefore P' is the middle point of A'B'. An Elementary Course in Synthetic Projective Geometry From the behavior at the asymptote, we can sketch the right side of the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z Little by little, the mountains will be worn down to a surface of less and less relief, approaching a plain as a hyperbola approaches its asymptote—a surface which W. M. Davis has called peneplain. Popular Science Monthly Oct, Nov, Dec, 1915 — Volume 86 For the following exercises, find the horizontal and vertical asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z With the stock market approaching the Y2K rollover like a hyperbola approaching its vertical asymptote, all talk of free software or open source as a political phenomenon falls by the wayside. Free as in Freedom: Richard Stallman's Crusade for Free Software Draw through the point of contact P of the tangent AB two lines, one parallel to one asymptote and the other parallel to the other. An Elementary Course in Synthetic Projective Geometry A function can have more than one vertical asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Are the difficulties that attend the doctrine of incommensurable quantities, of the angle of contact, of the asymptotes to curves, or the like, sufficient to make you hold out against mathematical demonstration? Three Dialogues Between Hylas and Philonous True or false: Every ratio of polynomials has vertical asymptotes. Calculus, Volume 1 2016-03-30T00:00:00Z The divine perfection is the asymptote of human life to which it is always striving, and always approaching, though it can only be reached in infinity. The Kingdom of God Is Within You Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. An Elementary Course in Synthetic Projective Geometry The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. Algebra and Trigonometry 2015-02-13T00:00:00Z The returned traveller entered the house, where he found awaiting him poor old Mrs. Martin, to whose earthly course death stood rather as the asymptote than as the end. Two on a Tower The Palais-Royal and I were two asymptotes bearing one towards the other, yet unable to meet. The Lily of the Valley There is therefore nothing more or less in the effect than in the group of causes; and the causal relation moves towards identity as towards its asymptote. A New Philosophy: Henri Bergson The tangents to a curve at its infinitely distant points are called its asymptotes if they pass through a finite part of the plane. An Elementary Course in Synthetic Projective Geometry Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. Algebra and Trigonometry 2015-02-13T00:00:00Z Those two lines are asymptotes, and will never meet. Father Goriot Note that this graph crosses the horizontal asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. Algebra and Trigonometry 2015-02-13T00:00:00Z The tangent AB may be fixed, and the tangent CD chosen arbitrarily; therefore The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area. An Elementary Course in Synthetic Projective Geometry TRY IT #1 Sketch the graph of State the domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. Algebra and Trigonometry 2015-02-13T00:00:00Z The denominator will be zero at indicating vertical asymptotes at these values. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #6 Find the vertical and horizontal asymptotes of the function: Algebra and Trigonometry 2015-02-13T00:00:00Z Given an asymptote and three finite points on the conic, to determine the conic. An Elementary Course in Synthetic Projective Geometry What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? Algebra and Trigonometry 2015-02-13T00:00:00Z The graph heads toward positive infinity as the inputs approach the asymptote on the right, so the graph will head toward positive infinity on the left as well. Algebra and Trigonometry 2015-02-13T00:00:00Z After passing through the x-intercepts, the graph will then level off toward an output of zero, as indicated by the horizontal asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z A function that levels off at a horizontal value has a horizontal asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z In particular, taking the point as the center of the conic, we find that conjugate diameters form a system of rays in involution, of which the asymptotes, if there are any, are the double rays. An Elementary Course in Synthetic Projective Geometry The left tail of the graph will approach the vertical asymptote and the right tail will increase slowly without bound. Algebra and Trigonometry 2015-02-13T00:00:00Z Find the slant asymptote of the rational function. Algebra and Trigonometry 2015-02-13T00:00:00Z Sketch the vertical asymptotes, which occur at The graph for this function is shown in Figure 11. Algebra and Trigonometry 2015-02-13T00:00:00Z The cotangent graph has vertical asymptotes at each value of where we show these in the graph below with dashed lines. Algebra and Trigonometry 2015-02-13T00:00:00Z Given three lines, two of which are asymptotes, to construct the conic. An Elementary Course in Synthetic Projective Geometry Include the key points and asymptotes on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z The vertical asymptotes are and The graph is shown in Figure 15. Algebra and Trigonometry 2015-02-13T00:00:00Z The zero of this factor, is the vertical asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the slant asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z The fourth is the line at infinity which joins the point of contact of the asymptotes. An Elementary Course in Synthetic Projective Geometry Include the key points and asymptote on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, graph the transformation of Give the horizontal asymptote, the domain, and the range. Algebra and Trigonometry 2015-02-13T00:00:00Z A logarithmic curve is always concave away from its vertical asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Also, conjugate diameters are harmonic conjugates with respect to the asymptotes. An Elementary Course in Synthetic Projective Geometry Include the key points and asymptote on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z What are the intercepts and asymptotes of each function? Algebra and Trigonometry 2015-02-13T00:00:00Z The left tail of the graph will approach the asymptote and the right tail will increase without bound. Algebra and Trigonometry 2015-02-13T00:00:00Z Given an equation with the general form we can identify the vertical asymptote for the transformation. Algebra and Trigonometry 2015-02-13T00:00:00Z The cosecant graph has vertical asymptotes at each value of where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines. Algebra and Trigonometry 2015-02-13T00:00:00Z Include the key points and asymptote on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z Sketch the vertical asymptotes, which occur at and There is a local minimum at and a local maximum at Figure 9 shows the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z The excluded points of the domain follow the vertical asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Include the key points and asymptote on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #2 Graph State domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #4 Sketch the graph of State the domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #3 Graph State the domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Include the key points and asymptotes on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Algebra and Trigonometry 2015-02-13T00:00:00Z Include the key points and asymptote on the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #7 Sketch a graph of the function State the domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #8 Graph State the domain, range, and asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z Does the graph of a general logarithmic function have a horizontal asymptote? Algebra and Trigonometry 2015-02-13T00:00:00Z Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, state the domain and the vertical asymptote of the function. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, state the domain, vertical asymptote, and end behavior of the function. Algebra and Trigonometry 2015-02-13T00:00:00Z An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Vertical asymptotes occur at the zeros of such factors. Algebra and Trigonometry 2015-02-13T00:00:00Z Label the foci and asymptotes, and draw a smooth curve to form the hyperbola, as shown in Figure 8. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #4 Graph the hyperbola given by the equation Identify and label the vertices, co- vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z What can we conclude about a hyperbola if its asymptotes intersect at the origin? Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero: Algebra and Trigonometry 2015-02-13T00:00:00Z The asymptotes are at and and the three recommended reference points are and The graph is shown in Figure 3. Algebra and Trigonometry 2015-02-13T00:00:00Z Sketch three asymptotes at and We can use two reference points, the local maximum at and the local minimum at Figure 10 shows the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z Identify the period, the phase shift, the amplitude, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points. Algebra and Trigonometry 2015-02-13T00:00:00Z Neither nor are zeros of the numerator, so the two values indicate two vertical asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Every hyperbola also has two asymptotes that pass through its center. Algebra and Trigonometry 2015-02-13T00:00:00Z As a hyperbola recedes from the center, its branches approach these asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. Algebra and Trigonometry 2015-02-13T00:00:00Z Sketch and extend the diagonals of the central rectangle to show the asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z The graph in Figure 9 confirms the location of the two vertical asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z The central rectangle and asymptotes provide the framework needed to sketch an accurate graph of the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown in Figure 9. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #5 Graph the hyperbola given by the standard form of an equation Identify and label the center, vertices, co-vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the equations of the asymptotes for each hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z Note that this graph crosses the horizontal asymptote. Algebra and Trigonometry 2015-02-13T00:00:00Z |
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