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Next Back Tangent and Cotangent Graphs Reading and Drawing Tangent and Cotangent Graphs Some slides in this presentation contain animation. Slides will be more meaningful if you allow each slide to finish its presentation before moving to the next one.

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Next Back This is the graph for y = tan x. This is the graph for y = cot x.

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Next Back One definition for tangent is. Notice that the denominator is cos x. This indicates a relationship between a tangent graph and a cosine graph. This is the graph for y = cos x.

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Next Back To see how the cosine and tangent graphs are related, look at what happens when the graph for y = tan x is superimposed over y = cos x.

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Next Back In the diagram below, y = cos x is drawn in gray while y = tan x is drawn in black. Notice that the tangent graph has horizontal asymptotes (indicated by broken lines) everywhere the cosine graph touches the x-axis.

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Next Back One definition for cotangent is. Notice that the denominator is sin x. This indicates a relationship between a cotangent graph and a sine graph. This is the graph for y = sin x.

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Next Back To see how the sine and cotangent graphs are related, look at what happens when the graph for y = cot x is superimposed over y = sin x.

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Next Back In the diagram below, y = sin x is drawn in gray while y = cot x is drawn in black. Notice that the cotangent graph has horizontal asymptotes (indicated by broken lines) everywhere the sine graph touches the x-axis.

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Next Back y = tan x. y = cot x. For tangent and cotangent graphs, the distance between any two consecutive vertical asymptotes represents one complete period.

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Next Back y = tan x. y = cot x. One complete period is highlighted on each of these graphs. For both y = tan x and y = cot x, the period is π. (From the beginning of a cycle to the end of that cycle, the distance along the x-axis is π.)

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Next Back For y = tan x, there is no phase shift. The y-intercept is located at the point (0,0). We will call that point, the key point.

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Next Back A tangent graph has a phase shift if the key point is shifted to the left or to the right.

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Next Back For y = cot x, there is no phase shift. Y = cot x has a vertical asymptote located along the y-axis. We will call that asymptote, the key asymptote.

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Next Back A cotangent graph has a phase shift if the key asymptote is shifted to the left or to the right.

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Next Back y = a tan b (x - c). For a tangent graph which has no vertical shift, the equation for the graph can be written as For a cotangent graph which has no vertical shift, the equation for the graph can be written as y = a cot b (x - c). c indicates the phase shift, also known as the horizontal shift. a indicates whether the graph reflects about the x-axis. b affects the period.

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Next Back y = a tan b (x - c) y = a cot b (x - c) Unlike sine or cosine graphs, tangent and cotangent graphs have no maximum or minimum values. Their range is (-∞, ∞), so amplitude is not defined. However, it is important to determine whether a is positive or negative. When a is negative, the tangent or cotangent graph will “flip” or reflect about the x-axis.

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Next Back Notice the behavior of y = tan x. Notice what happens to each section of the graph as it nears its asymptotes. As each section nears the asymptote on its left, the y-values approach - ∞. As each section nears the asymptote on its right, the y-values approach + ∞.

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Next Back Notice what happens to each section of the graph as it nears its asymptotes. As each section nears the asymptote on its left, the y-values approach + ∞. As each section nears the asymptote on its right, the y-values approach - ∞. Notice the behavior of y = cot x.

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Next Back This is the graph for y = tan x. y = - tan x Consider the graph for y = - tan x In this equation a, the numerical coefficient for the tangent, is equal to -1. The fact that a is negative causes the graph to “flip” or reflect about the x-axis.

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Next Back This is the graph for y = cot x. y = - 2cot x Consider the graph for y = - 2 cot x In this equation a, the numerical coefficient for the cotangent, is equal to -2. The fact that a is negative causes the graph to “flip” or reflect about the x-axis.

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Next Back y = a tan b (x - c) y = a cot b (x - c) b affects the period of the tangent or cotangent graph. For tangent and cotangent graphs, the period can be determined by Conversely, when you already know the period of a tangent or cotangent graph, b can be determined by

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Next Back A complete period (including two consecutive vertical asymptotes) has been highlighted on the tangent graph below. The distance between the asymptotes in this graph is. Therefore, the period of this graph is also. For all tangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

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Next Back We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis. Use, the period of this tangent graph, to calculate b. An equation for this graph can be written as or.

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Next Back A complete period (including two consecutive vertical asymptotes) has been highlighted on the cotangent graph below. The distance between the asymptotes is. Therefore, the period of this graph is also. For all cotangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

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Next Back We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis. Use, the period of this cotangent graph, to calculate b. An equation for this graph can be written as or.

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Next Back y = tan x has no phase shift. We designated the y-intercept, located at (0,0), as the key point.

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Next Back y = cot x has no phase shift. We designated the vertical asymptote on the y-axis (at x = 0) as the key asymptote. x = 0

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Next Back If the key point on a tangent graph shifts to the left or to the right, or if the key asymptote on a cotangent graph shifts to the left or to the right, that horizontal shift is called a phase shift.

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Next Back y = a tan b (x - c) c indicates the phase shift of a tangent graph. For a tangent graph, the x-coordinate of the key point is c. For this graph, c = because the key point shifted spaces to the right. An equation for this graph can be written as.

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Next Back y = a cot b (x – c) c indicates the phase shift of a cotangent graph. For a cotangent graph, c is the value of x in the key vertical asymptote. For this graph, c = because the key asymptote shifted left to. An equation for this graph can be written as or

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Next Back Graphs whose equations can be written as a tangent function can also be written as a cotangent function. Given the graph above, it is possible to write an equation for the graph. We will look at how to write both a tangent equation that describes this graph and a cotangent equation that describes the graph. The tangent equation will be written as y = a tan b (x – c). The cotangent equation will be written as y = a cot b (x – c).

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Next Back For the tangent function, the values for a, b, and c must be determined. This tangent graph has reflected about the x-axis, so a must be negative. We will use a = -1. The period of the graph is. The key point did not shift, so the phase shift is 0. c = 0

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Next Back The tangent equation for this graph can be written as or.

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Next Back For the cotangent function, the values for a, b, and c must be determined. This cotangent graph has not reflected about the x-axis, so a must be positive. We will use a = 1. The period of the graph is. The key asymptote has shifted spaces to the right, so the phase shift is. Therefore,.

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Next Back The cotangent equation for this graph can be written as.

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Next Back It is important to be able to draw a tangent graph when you are given the corresponding equation. Consider the equation Begin by looking at a, b, and c.

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Next Back The negative sign here means that the tangent graph reflects or “flips” about the x-axis. The graph will look like this.

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Next Back b = 3 Use b to calculate the period. Remember that the period is the distance between vertical asymptotes.

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Next Back This phase shift means the key point has shifted spaces to the right. It’s x-coordinate is. Also, notice that the key point is an x-intercept.

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Next Back The period is ; half of the period is. Therefore, the distance between the x-intercept and the asymptotes on either side is. Since the key point, an x-intercept, is exactly halfway between two vertical asymptotes, the distance from this x-intercept to the vertical asymptote on either side is equal to half of the period.

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Next Back We can use half of the period to figure out the labels for vertical asymptotes and x-intercepts on the graph. Since we already determined that there is an x-intercept at, we can add half of the period to find the vertical asymptote to the right of this x-intercept. x-intercept Half of the period Vertical asymptote

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Next Back Continue to add or subtract half of the period,, to determine the labels for additional x-intercepts and vertical asymptotes. Vertical asymptote Half of the period x-intercept

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Next Back It is important to be able to draw a cotangent graph when you are given the corresponding equation. Consider the equation Begin by looking at a, b, and c.

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Next Back The positive sign here means that the cotangent graph does not reflect or “flip” about the x-axis. The graph will look like this.

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Next Back b = 4 Use b to calculate the period. Remember that the period is the distance between vertical asymptotes.

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Next Back This phase shift means the key asymptote has shifted spaces to the left. The equation for this key asymptote is.

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Next Back The period is ; half of the period is. Therefore, the distance between asymptotes and their adjacent x-intercepts is. This information can be used to label asymptotes and x-intercepts. The distance from an asymptote to the x-intercepts on either side of it is equal to half of the period.

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Next Back Sometimes a tangent or cotangent graph may be shifted up or down. This is called a vertical shift. y = a tan b (x - c) +d. The equation for a tangent graph with a vertical shift can be written as The equation for a cotangent graph with a vertical shift can be written as y = a cot b (x - c) +d. In both of these equations, d represents the vertical shift.

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Next Back A good strategy for graphing a tangent or cotangent function that has a vertical shift: Graph the function without the vertical shift Shift the graph up or down d units. Consider the graph for. The equation is in the form where “d” equals 3, so the vertical shift is 3. The graph of was drawn in the previous example.

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Next Back To draw, begin with the graph for. Draw a new horizontal axis at y = 3. Then shift the graph up 3 units. 3 The graph now represents.

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Next Back This concludes Tangent and Cotangent Graphs.

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