# Logarithmic Functions Math@TutorVista com

In this section, you will:In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of 5%, compounded continuously. We already know that the balance in our account for any year t can be found with the equation A=2500 e 0.05t . But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? [link] shows this point on the logarithmic graph.In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.Recall that the exponential function is defined as y= b x for any real number x and constant b>0, whereisisIn the last section we learned that the logarithmic function y= log b ( x ) is the inverse of the exponential function y= b x . So, as inverse functions:is the range ofis the domain ofTransformations of the parent function y= log b ( x ) behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.In Graphs of Exponential Functions we saw that certain transformations can change the range of y= b x . Similarly, applying transformations to the parent function y= log b ( x ) can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.For example, consider f(x)= log 4 ( 2x−3 ). This function is defined for any values of x such that the argument, in this case 2x−3, is greater than zero. To find the domain, we set up an inequality and solve for x: In interval notation, the domain of f(x)= log 4 ( 2x−3 ) is ( 1.5,∞ ). Given a logarithmic function, identify the domain. What is the domain of f(x)= log 2 (x+3)? The logarithmic function is defined only when the input is positive, so this function is defined when x+3>0. Solving this inequality,The domain of f(x)= log 2 (x+3) is ( −3,∞ ). What is the domain of f(x)= log 5 (x−2)+1? What is the domain of f(x)=log(5−2x)? The logarithmic function is defined only when the input is positive, so this function is defined when 5–2x>0. Solving this inequality,The domain of f(x)=log(5−2x) is ( –∞, 5 2 ). What is the domain of f(x)=log(x−5)+2? Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function y= log b ( x ) along with all its transformations: shifts, stretches, compressions, and reflections.We begin with the parent function y= log b ( x ). Because every logarithmic function of this form is the inverse of an exponential function with the form y= b x , their graphs will be reflections of each other across the line y=x. To illustrate this, we can observe the relationship between the input and output values of y= 2 x and its equivalent x= log 2 (y) in [link].| </math></strong> | −3 | | </math></strong> | 1 8 | | </math></strong> | −3 Using the inputs and outputs from [link], we can build another table to observe the relationship between points on the graphs of the inverse functions f(x)= 2 x and g(x)= log 2 (x). See [link].| </math></strong> | ( −3, 1 8 ) | | </math></strong> | ( 1 8 ,−3 ) As we’d expect, the x- and y-coordinates are reversed for the inverse functions. [link] shows the graph of f and g. Observe the following from the graph:has a y-intercept atandhas an x- intercept atis the same as the range ofis the same as the domain ofFor any real number x and constant b>0, we can see the following characteristics in the graph of f(x)= log b ( x ): and key pointSee [link].[link] shows how changing the base b in f(x)= log b ( x ) can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function ln( x ) has base e≈2.718.) **Given a logarithmic function with the form f(x)= log b ( x ), graph the function.**the range,and the vertical asymptote,Graph f(x)= log 5 ( x ). State the domain, range, and asymptote.Before graphing, identify the behavior and key points for the graph.is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptoteand the right tail will increase slowly without bound.is on the graph.The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Graph f(x)= log 1 5 (x). State the domain, range, and asymptote. The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function y= log b ( x ) without loss of shape.When a constant c is added to the input of the parent function f(x)=lo g b (x), the result is a horizontal shift c units in the opposite direction of the sign on c. To visualize horizontal shifts, we can observe the general graph of the parent function f(x)= log b ( x ) and for c>0 alongside the shift left, g(x)= log b ( x+c ), and the shift right, h(x)= log b ( x−c ). See [link].For any constant c, the function f(x)= log b ( x+c ) leftunits ifrightunits if**Given a logarithmic function with the form f(x)= log b ( x+c ), graph the translation.**shift the graph ofleftunits.shift the graph ofrightunits.from thecoordinate.the range isand the vertical asymptote isSketch the horizontal shift f(x)= log 3 (x−2) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.Since the function is f(x)= log 3 (x−2), we notice x+( −2 )=x–2. Thus c=−2, so c<0. This means we will shift the function f(x)= log 3 (x) right 2 units.The vertical asymptote is x=−(−2) or x=2. Consider the three key points from the parent function, ( 1 3 ,−1 ), and ( 3,1 ). The new coordinates are found by adding 2 to the x coordinates.Label the points ( 7 3 ,−1 ), and ( 5,1 ). The domain is ( 2,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=2. Sketch a graph of f(x)= log 3 (x+4) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote. The domain is ( −4,∞ ), the range ( −∞,∞ ), and the asymptote x=–4. When a constant d is added to the parent function f(x)= log b ( x ), the result is a vertical shift d units in the direction of the sign on d. To visualize vertical shifts, we can observe the general graph of the parent function f(x)= log b ( x ) alongside the shift up, g(x)= log b ( x )+d and the shift down, h(x)= log b ( x )−d. See [link].For any constant d, the function f(x)= log b ( x )+d upunits ifdownunits if**Given a logarithmic function with the form f(x)= log b ( x )+d, graph the translation.**shift the graph ofupunits.shift the graph ofdownunits.to thecoordinate.the range isand the vertical asymptote isSketch a graph of f(x)= log 3 (x)−2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Since the function is f(x)= log 3 (x)−2, we will notice d=–2. Thus d<0. This means we will shift the function f(x)= log 3 (x) down 2 units.The vertical asymptote is x=0. Consider the three key points from the parent function, ( 1 3 ,−1 ), and ( 3,1 ). The new coordinates are found by subtracting 2 from the y coordinates.Label the points ( 1 3 ,−3 ), and ( 3,−1 ). The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Sketch a graph of f(x)= log 2 (x)+2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote. The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. When the parent function f(x)= log b ( x ) is multiplied by a constant a>0, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set a>1 and observe the general graph of the parent function f(x)= log b ( x ) alongside the vertical stretch, g(x)=a log b ( x ) and the vertical compression, h(x)= 1 a log b ( x ). See [link].For any constant a>1, the function f(x)=a log b ( x ) vertically by a factor ofifvertically by a factor ofif**Given a logarithmic function with the form f(x)=a log b ( x ), graph the translation.**the graph ofis stretched by a factor ofunits.the graph ofis compressed by a factor ofunits.coordinates bythe range isand the vertical asymptote isSketch a graph of f(x)=2 log 4 (x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Since the function is f(x)=2 log 4 (x), we will notice a=2. This means we will stretch the function f(x)= log 4 (x) by a factor of 2.The vertical asymptote is x=0. Consider the three key points from the parent function, ( 1 4 ,−1 ), and ( 4,1 ). The new coordinates are found by multiplying the y coordinates by 2.Label the points ( 1 4 ,−2 ), and ( 4,2 ). The domain is ( 0, ∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. See [link].The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Sketch a graph of f(x)= 1 2 log 4 (x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote. The domain is ( 0,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Sketch a graph of f(x)=5log(x+2). State the domain, range, and asymptote.Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link]. The vertical asymptote will be shifted to x=−2. The x-intercept will be (−1,0). The domain will be ( −2,∞ ). Two points will help give the shape of the graph: (−1,0) and (8,5). We chose x=8 as the x-coordinate of one point to graph because when x=8, the base of the common logarithm.The domain is ( −2,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=−2. Sketch a graph of the function f(x)=3log(x−2)+1. State the domain, range, and asymptote. The domain is ( 2,∞ ), the range is ( −∞,∞ ), and the vertical asymptote is x=2. When the parent function f(x)= log b ( x ) is multiplied by −1, the result is a reflection about the x-axis. When the input is multiplied by −1, the result is a reflection about the y-axis. To visualize reflections, we restrict b>1, and observe the general graph of the parent function f(x)= log b ( x ) alongside the reflection about the x-axis, g(x)= −log b ( x ) and the reflection about the y-axis, h(x)= log b ( −x ). The function f(x)= −log b ( x ) about the x-axis.range,and vertical asymptote,which are unchanged from the parent function.The function f(x)= log b ( −x ) about the y-axis.and vertical asymptote,which are unchanged from the parent function.**Given a logarithmic function with the parent function f(x)= log b ( x ), graph a translation.**Sketch a graph of f(x)=log(−x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Before graphing f(x)=log(−x), identify the behavior and key points for the graph.is greater than one, we know that the parent function is increasing. Since the input value is multiplied byis a reflection of the parent graph about the y-axis. Thus,will be decreasing asmoves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptoteThe domain is ( −∞,0 ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Graph f(x)=−log(−x). State the domain, range, and asymptote. The domain is ( −∞,0 ), the range is ( −∞,∞ ), and the vertical asymptote is x=0. Given a logarithmic equation, use a graphing calculator to approximate solutions.we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value offor the point(s) of intersection.Solve 4ln( x )+1=−2ln( x−1 ) graphically. Round to the nearest thousandth.Press [Y=] and enter 4ln( x )+1 next to Y1=. Then enter −2ln( x−1 ) next to Y2=. For a window, use the values 0 to 5 for x and –10 to 10 for y. Press [GRAPH]. The graphs should intersect somewhere a little to right of x=1. For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?) So, to the nearest thousandth, x≈1.339. Solve 5log( x+2 )=4−log( x ) graphically. Round to the nearest thousandth.Now that we have worked with each type of translation for the logarithmic function, we can summarize each in [link] to arrive at the general equation for translating exponential functions.All translations of the parent logarithmic function, y= log b ( x ), have the formwhere the parent function, y= log b ( x ),b>1, isunits.units.ififFor f( x )=log( −x ), the graph of the parent function is reflected about the y-axis.What is the vertical asymptote of f(x)=−2 log 3 (x+4)+5? The vertical asymptote is at x=−4. The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to x=−4. What is the vertical asymptote of f(x)=3+ln(x−1)? Find a possible equation for the common logarithmic function graphed in [link].This graph has a vertical asymptote at x=–2 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:It appears the graph passes through the points ( –1,1 ) and ( 2,–1 ). Substituting ( –1,1 ), Next, substituting in ( 2,–1 ) ,This gives us the equation f(x)=– 2 log(4) log(x+2)+1. We can verify this answer by comparing the function values in [link] with the points on the graph in [link].Give the equation of the natural logarithm graphed in [link].Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?Yes, if we know the function is a general logarithmic function. For example, look at the graph in [link]. The graph approaches</math>(or thereabouts) more and more closely, so</math>is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right,</math>The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as</math>and as</math></em>Access these online resources for additional instruction and practice with graphing logarithms.See [link] and [link]has an x-intercept atdomainrangevertical asymptoteandthe function is increasing.the function is decreasing.See [link].shifts the parent functionhorizontallyunits ifunits ifSee [link].shifts the parent functionverticallyunits ifunits ifSee [link].the equationvertically by a factor ofifvertically by a factor ofifSee [link] and [link].is multiplied bythe result is a reflection about the x-axis. When the input is multiplied bythe result is a reflection about the y-axis.represents a reflection of the parent function about the x-axis.represents a reflection of the parent function about the y-axis.See [link].See [link].we can identify the vertical asymptotefor the transformation. See [link].we can write the equation of a logarithmic function given its graph. See [link].The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?Since the functions are inverses, their graphs are mirror images about the line y=x. So for every point (a,b) on the graph of a logarithmic function, there is a corresponding point (b,a) on the graph of its inverse exponential function.What type(s) of translation(s), if any, affect the range of a logarithmic function?What type(s) of translation(s), if any, affect the domain of a logarithmic function?Shifting the function right or left and reflecting the function about the y-axis will affect its domain.Consider the general logarithmic function f(x)= log b ( x ). Why can’t x be zero?Does the graph of a general logarithmic function have a horizontal asymptote? Explain.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.For the following exercises, state the domain and range of the function.Domain: ( −∞, 1 2 ); Range: ( −∞,∞ ) Domain: ( − 17 4 ,∞ ); Range: ( −∞,∞ ) For the following exercises, state the domain and the vertical asymptote of the function.Domain: ( 5,∞ ); Vertical asymptote: x=5 Domain: ( − 1 3 ,∞ ); Vertical asymptote: x=− 1 3 Domain: ( −3,∞ ); Vertical asymptote: x=−3 For the following exercises, state the domain, vertical asymptote, and end behavior of the function.Domain: ( 3 7 , ∞ ); * * *Vertical asymptote: x= 3 7 ; End behavior: as x→ ( 3 7 ) + ,f(x)→−∞ and as x→∞,f(x)→∞ Domain: ( −3,∞ ) ; Vertical asymptote: x=−3 ; * * *End behavior: as x→− 3 + , f(x)→−∞ and as x→∞, f(x)→∞ For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.Domain: ( 1,∞ ); Range: ( −∞,∞ ); Vertical asymptote: x=1; x-intercept: ( 5 4 ,0 ); y-intercept: DNEDomain: ( −∞,0 ); Range: ( −∞,∞ ); Vertical asymptote: x=0; x-intercept: ( − e 2 ,0 ); y-intercept: DNEDomain: ( 0,∞ ); Range: ( −∞,∞ ); Vertical asymptote: x=0; x-intercept: ( e 3 ,0 ); y-intercept: DNEFor the following exercises, match each function in [link] with the letter corresponding to its graph.BCFor the following exercises, match each function in [link] with the letter corresponding to its graph.BCFor the following exercises, sketch the graphs of each pair of functions on the same axis.and g(x)= 10 x and g(x)= log 1 2 (x) and g(x)=ln(x) and g(x)=ln(x) For the following exercises, match each function in [link] with the letter corresponding to its graph.CFor the following exercises, sketch the graph of the indicated function.For the following exercises, write a logarithmic equation corresponding to the graph shown.Use y= log 2 (x) as the parent function. Use f(x)= log 3 (x) as the parent function. Use f(x)= log 4 (x) as the parent function. Use f(x)= log 5 (x) as the parent function. For the following exercises, use a graphing calculator to find approximate solutions to each equation.Let b be any positive real number such that b≠1. What must log b 1 be equal to? Verify the result.Explore and discuss the graphs of f(x)= log 1 2 ( x ) and g(x)=− log 2 ( x ). Make a conjecture based on the result.The graphs of f(x)= log 1 2 ( x ) and g(x)=− log 2 ( x ) appear to be the same; Conjecture: for any positive base b≠1, Prove the conjecture made in the previous exercise.What is the domain of the function f(x)=ln( x+2 x−4 )? Discuss the result.Recall that the argument of a logarithmic function must be positive, so we determine where x+2 x−4 >0 . From the graph of the function f( x )= x+2 x−4 , note that the graph lies above the x-axis on the interval ( −∞,−2 ) and again to the right of the vertical asymptote, that is ( 4,∞ ). Therefore, the domain is ( −∞,−2 )∪( 4,∞ ). Use properties of exponents to find the x-intercepts of the function f(x)=log( x 2 +4x+4 ) algebraically. Show the steps for solving, and then verify the result by graphing the function.You can also download for free at http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@Attribution:

Download presentationWe think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!Buttons: Presentation is loading. Please wait.Published byGretchen Sailor Modified over 2 years ago 1 Graph of Exponential FunctionsChapter 3 Lesson G { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/1/Graph+of+Exponential+Functions.jpg", "name": "Graph of Exponential Functions", "description": "Chapter 3. Lesson G.", "width": "800" } 2 Exponential FunctionsThe general exponential function has form f(x) = a^x where a>0, a≠1 y=0 is a horizontal asymptote { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/2/Exponential+Functions.jpg", "name": "Exponential Functions", "description": "The general exponential function has form f(x) = a^x where a>0, a≠1. y=0 is a horizontal asymptote.", "width": "800" } 3 Example Graph y= 2^x { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/3/Example+Graph+y%3D+2%5Ex.jpg", "name": "Example Graph y= 2^x", "description": "Example Graph y= 2^x", "width": "800" } 4 General Exponential Functiony= a x b ^(x-c) +d b controls how steeply the graph increases or decreases c controls horizontal translation D controls vertical translation and y=d is the equation of the horizontal asymptote { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/4/General+Exponential+Function.jpg", "name": "General Exponential Function", "description": "y= a x b ^(x-c) +d. b controls how steeply the graph increases or decreases. c controls horizontal translation. D controls vertical translation and y=d is the equation of the horizontal asymptote.", "width": "800" } 5 General Exponential FunctionIf a>0, b>1 If a>0, 0<b<1 If a<0, b>1 If a<0, 0<b<1 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/5/General+Exponential+Function.jpg", "name": "General Exponential Function", "description": "If a>0, b>1. If a>0, 00, 0 6 Horizontal Asymptotesy= a x b ^(x-c) +d In this function y=d is the horizontal asymptote Using these you can obtain a reasonably accurate sketch of the graph Horizontal asymptote Y-intercept Two other points, say when x=2, x=-2 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/6/Horizontal+Asymptotes.jpg", "name": "Horizontal Asymptotes", "description": "y= a x b ^(x-c) +d. In this function y=d is the horizontal asymptote. Using these you can obtain a reasonably accurate sketch of the graph. Horizontal asymptote. Y-intercept. Two other points, say when x=2, x=-2.", "width": "800" } 7 Example { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/7/Example.jpg", "name": "Example", "description": "Example", "width": "800" } 8 Homework Page #1-5 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/2559230/9/images/8/Homework+Page+77-78+%231-5.jpg", "name": "Homework Page 77-78 #1-5", "description": "Homework Page 77-78 #1-5", "width": "800" } Powerpoint template free download ppt on pollution Mp ppt online application form Ppt on addition for class 2 Ppt on kingdom monera organisms Ppt on fibonacci numbers nature Ppt on suspension type insulators of electricity Ppt on security features of atm cards Ppt on op amp circuits diagrams Ppt on inhabiting other planets in other galaxies Ppt on meeting skills pdf Graphs of Exponential and Logarithmic FunctionsMath – Exponential FunctionsDo Now: State the domain of the function.. Academy Algebra II 7.1, 7.2: Graph Exponential Growth and Decay Functions HW: p.482 (6, 10, even), p.489.Translations Translations and Getting Ready for Reflections by Graphing Horizontal and Vertical Lines.6.3 Logarithmic Functions. Change exponential expression into an equivalent logarithmic expression. Change logarithmic expression into an equivalent.What is the symmetry? f(x)= x 3 –x.Asymptotes Objective: -Be able to find vertical and horizontal asymptotes.Equation of a line y = m x + bMath 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2.General Form and Graph for an Exponential Function.Lesson 8.1. Exponential Function: a function that involves the expression b x where the base b is a positive number other than 1. Asymptote: a line.Create a table and Graph:. Reflect: Continued x-intercept: y-intercept: Asymptotes: xy -31/3 -21/2 1 -1/22 xy 1/ /2 3-1/3.Double Jeopardy $200 Miscellaneous Facts Solving Logs with Properties Solving Log Equations Solving Exponential Equations Graphs of Logs $400 $600 $800.6.2 Exponential Functions. An exponential function is a function of the form where a is a positive real number (a > 0) and. The domain of f is the set.8-2 Properties of Exponential Functions. The function f(x) = b x is the parent of a family of exponential functions for each value of b. The factor a.Lesson 3.2 Read: Pages Handout 1-49 (ODD), 55, 59, 63, 68, (ODD)Lesson 3.1 Read: Pages Handout #1-11 (ODD), (ODD), (EOO), (ODD), (EOO)Notes Over 9.2 Graphing a Rational Function The graph of a has the following characteristics. Horizontal asymptotes: center: Then plot 2 points to the.Graph Y-Intercept =(0,2) Horizontal Asymptote X-Axis (y = 0) Domain: All Real Numbers Range: y > 0.6.7 Graphing Absolute Value Equations. Vertical Translations Below are the graphs of y = | x | and y = | x | + 2. Describe how the graphs are the same. Similar presentations © 2017 SlidePlayer.com Inc. All rights reserved.