(b)  How high is the highest point of the roller coaster? In this case, there's a –2.5 multiplied directly onto the tangent. 2 a Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Math Homework. To get the high point, take the middle (average) of \(\displaystyle -\frac{\pi }{6}\text{ and }\frac{\pi }{2}\), which is \(\displaystyle \frac{\pi }{6}\). 4.9/5.0 Satisfaction Rating over the last 100,000 sessions. sin Now we have \(\displaystyle y=6\cos \frac{\pi }{{800}}\left( {x-c} \right)-4\). We see that the graph repeats itself from when \(\displaystyle x=-\frac{\pi }{{10}}\) to when \(\displaystyle x=\frac{{3\pi }}{{10}}\), so the new period is \(\displaystyle \frac{{3\pi }}{{10}}-\left( {-\frac{\pi }{{10}}} \right)=\frac{{4\pi }}{{10}}=\frac{{2\pi }}{5}\). Domain:  \(\left( {-\infty ,\infty } \right)\)     Range: \(\left[ {-4,\,\,4} \right]\). The \(y\)’s stay the same; subtract \(\displaystyle \frac{\pi }{4}\) from the \(x\) values (we do the opposite math when working with the \(x\)’s). How high is the coaster at \(x=\), (c)  How long (horizontally) is the roller coaster when the track is, The weight on a long spring bounces up and down sinusoidally with time. Put the trig function in the \(y=a\csc b\left( {x-c} \right)+d\,\,\,\text{or}\,\,\,y=a\sec b\left( {x-c} \right)+d\) format. a If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).   Drawing Transformed Graphs for Sin and Cos. In other words, if you shift the function by half of a period, then the resulting function is the opposite the original function. As of 4/27/18. Now we have \(y=6\cos b\left( {x-c} \right)-4\). This value is approximately .63 seconds: Let’s do one more, where we’ll use a sin function: A tsunami or tidal wave is an ocean wave caused by an earthquake. Use the equation \(y=a\sin b\left( {x-c} \right)+d\), where \(\left| a \right|\) is the amplitude, \(\displaystyle b=\frac{{2\pi }}{{\text{period}}}\), \(c\) is the phase shift, and \(d\) is the vertical shift. | Since the period is \(6\pi \), and the phase shift is right \(\displaystyle \frac{\pi }{3}\), so far we have \(\require {cancel} \displaystyle y=a\sin \left( {\frac{{{{{\cancel{{2\pi }}}}^{1}}}}{{{{{\cancel{{6\pi }}}}^{3}}}}\left( {x-\frac{\pi }{3}} \right)} \right)+d\). Also, sometimes, the graphs will be “upside down” which means you might need to reflect the sin or cos (using a negative coefficient). = You can use the same steps to see that when the roller coaster track is 15 feet above the ground, the roller coaster is 100 feet from the beginning point. ) For the t-chart, remember that for the \(x\), we do the opposite math (\(3x\) instead of \(\displaystyle \frac{1}{3}x\)). Domain: \(\left( {-\infty ,\infty } \right)\)      Range: \(\left[ {-1,3} \right]\). Amplitude = | a | Let b be a real number. The graphs of `tan x`, `cot x`, `sec x` and `csc x` are not as common as the sine and cosine curves that we met earlier in this chapter. Here are some examples; note that answers may vary: Uh oh – more word problems! However, they do occur in engineering and science problems. \(\boldsymbol{d=}\) the vertical shift of the graph (sometimes called a bias). To get the asymptotes, start with the regular tan asymptotes, set to the new tan argument, and solve for \(x\): \(\displaystyle \frac{\pi }{2}x+\pi =\frac{\pi }{2}+\pi k;\,\,\,\,\frac{\pi }{2}x=\left( {\frac{\pi }{2}-\pi } \right)+\pi k;\,\,\,\frac{\pi }{2}x=-\frac{\pi }{2}+\pi k;\,\,\,x=-1+2k\). Since the graph isn’t shifted to the left or right from the \(y\)-axis, there is no phase shift: \(c=0\). Now we have \(y=2\sin b\left( {x-c} \right)+1\).   Without a shift, \(a=1\) (thus the sin and cos graphs go from \(y=-1\) to \(y=1\), with the middle at \(y=0\). Note that some teachers may have you use a method that looks at the zeros of the sin and cosine functions. A periodic function is a function whose graph repeats itself identically from left to right. = These changes will affect the \(y\) values only. The time taken for the particle to complete one oscilation, that is, the time taken for the particle to move from its starting position and return to its original position is known as the period. pxx = periodogram(x) returns the periodogram power spectral density (PSD) estimate, pxx, of the input signal, x, found using a rectangular window.When x is a vector, it is treated as a single channel.   ( How high is the coaster at \(x=\) 15 feet? Do It Faster, Learn It Better. To get the middle of the function, or the vertical shift (\(d\)), we add the amplitude to the lowest \(y\) value: \(30+10=40\). Here are some examples of drawing transformed trig graphs, first with the sin function, and then the cos (the rest of the trig functions will be addressed later). A 1.75−kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured in metres and time in seconds. A nonzero constant P for which this is the case is called a period of the function. of the graph, let’s see how close it is to the \(y\)-axis \((x=0)\); this value is \(c\). Sinusoids are quite useful in many scientific fields; sine waves are everywhere! The amplitude of the function is There is one small trick to remember about A, B, C, and D. We see that the graph repeats itself from when \(x=0\) to when \(x=\pi \), so the new period is \(\pi -0=\pi \).   x a cos Asymptotes: \(\displaystyle x=-\frac{\pi }{4}+\pi k,\,\,k\in \,\text{Int}\), Domain: \(\displaystyle x\ne -\frac{\pi }{4}+\pi k\)      Range: \(\left( {-\infty ,-3} \right]\cup \left[ {5,\infty } \right)\), Period: \(2\pi \)        Vertical Stretch:  4. See the screens on the left to see how we can check a complete revolution of the graph in a graphing calculator – looks good!         ) From here we go, The intermediate points will be halfway in between each, Find the amplitude (\(a\)) of the transformed function by subtracting the bottom \(y\) value from the top \(y\) value, and then dividing by, To get \(d\), or the vertical shift of the function, we add the amplitude to the bottom \(y\) value.   The graph is \(y=2\sin 2x+1\).   Multiplying and Dividing, including GCF and LCM, Powers, Exponents, Radicals (Roots), and Scientific Notation, Introduction to Statistics and Probability, Types of Numbers and Algebraic Properties, Coordinate System and Graphing Lines including Inequalities, Direct, Inverse, Joint and Combined Variation, Introduction to the Graphing Display Calculator (GDC), Systems of Linear Equations and Word Problems, Algebraic Functions, including Domain and Range, Scatter Plots, Correlation, and Regression, Solving Quadratics by Factoring and Completing the Square, Solving Absolute Value Equations and Inequalities, Solving Radical Equations and Inequalities, Advanced Functions: Compositions, Even and Odd, and Extrema, The Matrix and Solving Systems with Matrices, Rational Functions, Equations and Inequalities, Graphing Rational Functions, including Asymptotes, Graphing and Finding Roots of Polynomial Functions, Solving Systems using Reduced Row Echelon Form, Conics: Circles, Parabolas, Ellipses, and Hyperbolas, Linear and Angular Speeds, Area of Sectors, and Length of Arcs, Law of Sines and Cosines, and Areas of Triangles, Introduction to Calculus and Study Guides, Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Rules, Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change, Implicit Differentiation and Related Rates, Differentials, Linear Approximation and Error Propagation, Exponential and Logarithmic Differentiation, Derivatives and Integrals of Inverse Trig Functions, Antiderivatives and Indefinite Integration, including Trig Integration, Riemann Sums and Area by Limit Definition, Applications of Integration: Area and Volume, T-Charts for the Six Trigonometric Functions. Here’s a general formula in order to transform a sin or cos function, as well as the  remaining four trig functions.     • Display both channels as a function of time. State the phase shift and vertical translation, if applicable. Notice that it’s on the inside of the parentheses; this shifts a graph horizontally and when \(c\) is subtracted from \(\boldsymbol{x}\), it shifts to the right (opposite of what we’d think; we saw this with non-trig transformations). Note that in the WINDOW screen, we can use \(\pi \) when we input the, Since the top of the graph is close to the \(y\)-axis, we will use the positive. You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic. . Find the period of the function which is the horizontal distance for the function to repeat.   Range:  \(\left( {-\infty ,\,\,\infty } \right)\). (a) What is the amplitude, frequency, angular frequency, and period of this motion? cos π represents half the distance between the maximum and minimum values of the function. The new period = \(\displaystyle \frac{\pi }{b}=\frac{\pi }{{\frac{1}{2}}}=2\pi \). If asked for asymptotes of transformed functions, you’ll perform the same transformations on them as you would the \(x\) values of the graph. Each describes a separate parameter in the most general solution of the wave equation. In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: → = − →, where k is a positive constant..   Amplitude   So half a revolution is \(900-100=800\), so a complete revolution is \(1600\). Here are the steps to do this; examples will follow. The basic function has an amplitude of one.   We are already given the amplitude (12 meters), vertical shift (normal depth is at 10 meters), and period (20 minutes), so \(\displaystyle b=\frac{2\pi }{\text{new period}}=\frac{2\pi }{20}=\frac{\pi }{10}\). : \(2-\left( {-10} \right)=12\div 2=6\). You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain, Range, Period and Amplitude for each graph. (c)  The approximate distance from the ground when the clock is at \(t=0\) seconds is  \(\displaystyle y=10\cos \frac{10\pi }{13}\left( 0-.2 \right)+40=48.85\) cm. There is no horizontal phase shift, so the sinusoidal function is \(\displaystyle y=-12\sin \frac{\pi }{10}x+10\).   b π If there is a negative sign before the \(a\), the graph is flipped across the \(\boldsymbol{x}\)-axis. We can use 5 key points for a whole period of a graph. x And now that you know how to transform sin and cos functions, that’s really all we’re doing here. π The new period = \(\displaystyle \frac{{2\pi }}{b}=\frac{{2\pi }}{5}\).   The new period = \(\displaystyle \frac{{2\pi }}{b}=\frac{{2\pi }}{{\frac{1}{3}}}=6\pi \). See below for clarification. 5 We can plot the following points and draw the graph: Now for this graph, we will use the sin function since the middle of the function goes through the \(y\)-axis (\(x=0\)). \(\boldsymbol{c=}\) the horizontal shift or phase shift of the graph.   ( The period of the function is *See complete details for Better Score Guarantee. | (a)  Let’s get the equation with these steps: (b)  At the highest point, the roller coaster is 90 feet above the ground (\(\displaystyle y=50\cos \left( \frac{\pi }{150}*0 \right)+40=90\)). (d)   What would be the first positive value for the time when the weight is 45 cm above the ground?     (We could also have just taken the average of, To get the period of the graph, we know that the. Put the trig function in the \(y=a\tan b\left( {x-c} \right)+d\,\,\,\text{or}\,\,\,y=a\cot b\left( {x-c} \right)+d\) format. 2   ( Shift graph \(\displaystyle \frac{\pi }{4}\) to the left. Note that sometimes you’ll see the formula arranged differently; for example, with “\(a\)” being the vertical shift at the beginning. Also note that “undef” means the function is undefined for that \(x\) value; there is a vertical asymptote there. If asked for asymptotes of transformed functions, you’ll perform the same transformations on them as you would the \(x\) values of the graph. For the following exercises, graph one full period of each function, starting at x = 0. x = 0. We know the lowest point is at 5 minutes, and the period is 20 minutes, we can figure out that the highest point is at half the distance of the period (10 minutes) from that lowest point. When x is a matrix, the PSD is computed independently for each column and stored in the corresponding column of pxx. A function f is said to be periodic if, for some nonzero constant P, it is the case that (+) = ()for all values of x in the domain. Since the highest point is on the \(y\)-axis (\(x=0\)), there is no horizontal phase shift. a Asymptotes: \(\displaystyle x=\frac{\pi }{{10}}+\frac{\pi }{5}k,\,\,k\in \,\text{Int}\), Domain: \(\displaystyle x\ne \frac{\pi }{{10}}+\frac{\pi }{5}k\)       Range: \(\left( {-\infty ,-1} \right]\cup \left[ {7,\infty } \right)\), Period: \(\displaystyle \frac{{2\pi }}{5}\)        Vertical Stretch:  4. Here are the steps for sin and cos graphs: (Note that there’s an example of this here). Asymptotes: \(\displaystyle x=\frac{\pi }{8}+\frac{\pi }{4}k,\,\,k\in \,\text{Int}\), Domain: \(\displaystyle x\ne \frac{\pi }{8}+\frac{\pi }{4}k\)      Range: \(\left( {-\infty ,\infty } \right)\), Period: \(\displaystyle \frac{\pi }{4}\)       Vertical Stretch: 2, \(\displaystyle y=\frac{1}{2}\cot \left( {\frac{1}{2}x-\pi } \right)\), \(\displaystyle y=\frac{1}{2}\cot \left( {\frac{1}{2}\left( {x-2\pi } \right)} \right)\). Note also that when the original functions (like sin, cos, and tan) have 0’s as \(y\) values, their respective reciprocal functions are undefined at those points (because of division of 0).   |, Let   These aren’t too bad, once you get the hang of them. Vertical compression of \(\displaystyle \frac{1}{2}\).     Graph will be flipped because of the negative sign. Here are some examples of drawing transformed trig graphs, first with the sin function, and then the cos (the rest of the trig functions will be addressed later). First flip graph across the \(\boldsymbol {x}\)-axis and stretch by factor of 4. Period To get the middle of the function, or the vertical shift (\(d\)), we add the amplitude to the lowest \(y\) value:  \(4+8=12.\). be a real number. Note the window I used to match the graph of the roller coaster. Find the period and amplitude of To get \(b\), we first find the period of the graph by seeing how long it goes before repeating itself (we can subtract the two \(x\) values to get this new period). = (starting with the middle points) by going, Transformations of all Trig Functions without T-Charts. \(\begin{array}{l}y=a\sin b\left( {x-c} \right)+d\\\\\\y=a\cos b\left( {x-c} \right)+d\end{array}\), \(\begin{array}{l}y=a\csc b\left( {x-c} \right)+d\\y=a\sec b\left( {x-c} \right)+d\\y=a\tan b\left( {x-c} \right)+d\\y=a\cot b\left( {x-c} \right)+d\end{array}\). Solution: (a)  Let’s get the equation with these steps: (b)  When the clock reads 18 seconds, we can plug in 18 for \(x\) to get \(y\) (the distance from the ground): \(\displaystyle y=10\cos \frac{10\pi }{13}\left( 18-.2 \right)+40\), which is about 45.68 cm. Let’s just start with an example, and see the steps: A part of the track of a roller coaster has the shape of a sinusoidal function. Both the \(x\) values and \(y\) values are affected. Domain: \(\left( {-\infty ,\infty } \right)\)      Range: \(\left[ {-1,1} \right]\). If the absolute value is on the outside, like \(y=\left| {\sin x} \right|\), reflect all the \(y\) values across the \(x\)-axis, and for \(y=\sin \left| x \right|\), “erase” all the negative x values and reflect the positive \(y\) values across the \(y\)-axis: You may be asked to write trig function equations, given transformed graphs. y = 36 cos (6 pi x - 2). cos Let \(y=\) the height of the track (with \(y=0\) as the ground), and \(x=\) the number of feet horizontally, with \(x=0\) at the highest point of the track. Solution: Rewrite The amplitude is 2, the period is π and the phase shift is π/4 units to the left.   Understand these problems, and practice, practice, practice! Remember, again, like the sin and cos transformations, the \(\displaystyle \text{new period}=\frac{{2\pi }}{b}\). Thus the sinusoidal function is \(\displaystyle y=50\cos \frac{\pi }{150}x+40\). and After you hit GRAPH, you may have to use the TRACE button to get the cursor closer to the first point of intersection before you use intersect. It has no phase or vertical shifts, because it is centered on the origin. Let’s say the amplitude for this particular tsunami is 12 meters, it’s period is about 20 minutes, and it’s normal depth is 10 meters.     Definition. The amplitude of It has a period of pi. Period, Amplitude and Frequency. Domain: \(x\ne 2\pi k\)        Range: \(\left( {-\infty ,\infty } \right)\), Period: \(2\pi \)         Vertical Compression: \(\displaystyle \frac{1}{2}\). The 2 nd harmonic (n=2) has exactly two oscillations in one period, T=1, of the original function, and an amplitude of a 2 =0.1871. ... cos cos = ω +θ ... • Automatic amplitude measurement (preferably rms value) for each channel. = It’s easiest to put the function \(y=50\cos \frac{\pi }{{150}}x+40\) in the graphing calculator, and use the 2nd TRACE (CALC) value function to get these values (see the WINDOW you can use below). (Writing equations from trig functions other than sin and cos may be found here). Here are a few examples where we get the equations of trig functions other than sin and cos from graphs. .large-mobile-banner-1-multi{display:block !important;float:none;line-height:0px;margin-bottom:15px !important;margin-left:0px !important;margin-right:0px !important;margin-top:15px !important;min-height:250px;min-width:970px;text-align:center !important;}eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_4',134,'0','0']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_5',134,'0','1']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_6',134,'0','2']));For Practice: Use the Mathway widget below to try a Trig Transformation problem. \(\displaystyle y=-4\csc \left( {x+\frac{\pi }{4}} \right)+1\). a We’ll have a total of five points. Normally, \(c=0\). If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations … 1. | For tan and cot, since normally the period is \(\pi\), we have: \(\displaystyle b=\frac{\pi }{{\text{new period}}}\) and \(\displaystyle \text{new period}=\frac{\pi }{b}\). (c)  How long (horizontally) is the roller coaster when the track is 75 feet above the ground? This is after many bounces, as you could see if you graphed the function and made the window of \(x\) larger. The new period = \(\displaystyle \frac{\pi }{b}=\frac{\pi }{4}\). You are looking at a second hand on a clock and notice that when the clock reads, (b)   What would be the approximate distance from the ground when the clock reads, (c)   What is the approximate distance from the ground when the clock was at \(t=\), (d)   What would be the first positive value for the time when the weight is. Basic Sine Function Periodic Functions Definition, Period, Phase Shift, Amplitude, Vertical Shift. |. ” in order to input this fraction in fraction mode). Graphs of tan, cot, sec and csc. Sometimes it helps to remember that the sin graphs start in the middle of the graph, and the cos graphs start at the top of the graph. Note that in order to perform the transformations accurately and quickly, you must know your 6 trig functions graphs inside out!   Domain: \(\left( {-\infty ,\infty } \right)\)     Range: \(\left[ {-5,-3} \right]\), \(\displaystyle y=-2\cos \left( {\frac{1}{3}x-\frac{\pi }{6}} \right)+1\), \(\displaystyle y=-2\cos \left( {\frac{1}{3}\left( {x-\frac{\pi }{2}} \right)} \right)+1\).   2. To get the asymptotes, start with the regular csc asymptotes, set to the new csc argument, and solve for \(x\): \(\displaystyle 2x=\pi k;\,\,\,\,x=\frac{{\pi k}}{2}\,\,\,\text{or }\frac{\pi }{2}k\). “B” is the period, so you can elongate or shorten the period by changing that constant.   \(\displaystyle x-\frac{\pi }{4}\)   \(x\,\), \(\displaystyle -\frac{\pi }{4}\)      \(0\), \(\displaystyle \frac{{\pi }}{4}\)      \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{{3\pi }}{4}\)     \(\pi \), \(\displaystyle \frac{{5\pi }}{4}\)     \(\displaystyle \frac{3\pi }{2}\), \(\displaystyle \frac{{7\pi }}{4}\)     \(2\pi \), \(\displaystyle \frac{{\pi }}{4}\)     \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{{\pi }}{2}\)      \(\pi \), \(\displaystyle \frac{{3\pi }}{4}\)     \(\displaystyle \frac{3\pi }{2}\), \(\displaystyle 3x+\frac{\pi }{2}\)  \(x\), \(\displaystyle \frac{{3\pi }}{2}\)     \(0\), \(2\pi \)     \(\displaystyle \frac{{\pi }}{2}\), \(\displaystyle \frac{{7\pi }}{2}\)     \(\pi \), \(5\pi \)     \(\displaystyle \frac{{3\pi }}{2}\), \(\displaystyle \frac{{13\pi }}{2}\)    \(2\pi \), \(\displaystyle x-\frac{\pi }{4}\)     \(x\), \(\displaystyle -\frac{\pi }{4}\)     \(0\), \(\displaystyle \frac{\pi }{4}\)     \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{{5\pi }}{4}\)    \(\displaystyle \frac{{3\pi }}{2}\), \(\displaystyle \frac{1}{5}x\)      \(x\), \(\displaystyle \frac{\pi }{10}\)      \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{\pi }{{5}}\)       \(\pi \), \(\displaystyle \frac{{3\pi }}{{10}}\)      \(\displaystyle \frac{{3\pi }}{{2}}\), \(\displaystyle \frac{{2\pi }}{{5}}\)      \(2\pi \), \(\displaystyle -\frac{\pi }{8}\)    \(\displaystyle -\frac{\pi }{2}\), \(\displaystyle -\frac{\pi }{{16}}\)   \(\displaystyle -\frac{\pi }{4}\), \(\displaystyle \frac{\pi }{16}\)     \(\displaystyle \frac{\pi }{4}\), \(\displaystyle \frac{\pi }{8}\)      \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{5\pi }{2}\)     \(\displaystyle \frac{\pi }{4}\), \(3\pi \)     \(\displaystyle \frac{\pi }{2}\), \(\displaystyle \frac{7\pi }{2}\)    \(\displaystyle \frac{3\pi }{4}\), The graph is centered at \(y=1\), because of the vertical shift. y   Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors.   a (d)  Let’s use the graphing calculator to find the first positive value when the weight is 45 cm above the ground. Work backwards to make sure we get the correct characteristics; we do! Move graph to the left \(\displaystyle \frac{\pi }{4}\) units. b   For the t-chart, remember that for the \(x\), we do the opposite math (\(\displaystyle \frac{1}{2}x\) instead of \(2x\)). Now we have \(\displaystyle y=8\sin \left( {\frac{1}{3}\left( {x-\frac{\pi }{3}} \right)} \right)+d\). With sinusoidal applications, you’ll typically have to decide between using a sin graph or a cos graph. State the maximum and minimum y-values and their corresponding x-values on one period for x > 0. x > 0. To get the period, take the regular tan period of \(\pi\) and divide by \(\displaystyle \frac{\pi }{2}\); period is 2 (distance between asymptotes). (a)   Draw the graph that represents this situation, and write the sinusoidal equation that expresses the distance from the ground in terms of the numbers of seconds that has passed.   Up to date, curated data provided by Mathematica's ElementData function from Wolfram Research, Inc. Click here to buy a book, photographic periodic table poster, card deck, or 3D print based on the images you see here! situation. We can see that the graph repeats, See the screens on the left to see how we can check this half revolution of the graph in a graphing calculator – looks good! =   We can also see from the graph that the middle of the graph is at \(y=1\). Amplitude, frequency, wavenumber, and phase shift are properties of waves that govern their physical behavior.   The triangle wave has half-wave symmetry. =   y | The function then repeats the procedure for the tallest remaining peak and iterates until it runs out of peaks to consider.   Differential Equation of Oscillations. The Transformations of Trig Functions section covers: We learned how to transform Basic Parent Functions here in the Parent Functions and Transformations section.

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