A lot of times, you can just tell by looking at it, but sometimes you have to use a point or two. There are several ways to perform transformations of parent functions; I like to use t-charts, since they work consistently with ever function. Again, the “parent functions” assume that we have the simplest form of the function; in other words, the function either goes through the origin \(\left( {0,0} \right)\), or if it doesn’t go through the origin, it isn’t shifted in any way. (we do the “opposite” math with the “\(x\)”), Domain:  \(\left[ {-9,9} \right]\)     Range: \(\left[ {-10,2} \right]\), Transformation: \(\displaystyle f\left( {\left| x \right|+1} \right)-2\), \(y\) changes:  \(\displaystyle f\left( {\left| x \right|+1} \right)\color{blue}{{\underline{{-\text{ }2}}}}\). There are three types of transformations: translations, reflections, and dilations. This depends on the direction you want to transoform. Parent: Transformations: For problems 10 — 14, given the parent function and a description of the transformation, write the equation of the transformed function, f(x). ), (Do the “opposite” when change is inside the parentheses or underneath radical sign.). , we have \(a=-3\), \(\displaystyle b=\frac{1}{2}\,\,\text{or}\,\,.5\), \(h=-4\), and \(k=10\). The six most common graphs are shown in Figures 1a-1f. If you didn’t learn it this way, see IMPORTANT NOTE below. \(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\), Critical points: \(\displaystyle \left( {-1,-1} \right),\,\left( {0,0} \right),\,\left( {1,1} \right)\), \(y=\left| x \right|\) Sometimes the problem will indicate what parameters (\(a\), \(b\), and so on) to look for. Domain: \(\left( {-\infty ,\infty } \right)\)     Range: \(\left( {-\infty ,\,\infty } \right)\). Even when using t-charts, you must know the general shape of the parent functions in order to know how to transform them correctly! 13. Function Transformations . You may also be asked to perform a transformation of a function using a graph and individual points; in this case, you’ll probably be given the transformation in function notation. To do this, to get the transformed \(y\), multiply the \(y\) part of the point by –6 and then subtract 2. When looking at the equation of the transformed function, however, we have to be careful. what transformations have secured to the parent function to form g(x) = (3) 060 - 6 The parent function is State the transformations. We need to find \(a\); use the given point \((0,4)\):      \(\begin{align}y&=a\left( {\frac{1}{{x+2}}} \right)+3\\4&=a\left( {\frac{1}{{0+2}}} \right)+3\\1&=\frac{a}{2};\,\,\,a=2\end{align}\). Note: we could have also noticed that the graph goes over 1 and up 2 from the center of asymptotes, instead of over 1 and up 1 normally with \(\displaystyle y=\frac{1}{x}\). Find the domain and the range of the new function. So, you would have \(\displaystyle {\left( {x,\,y} \right)\to \left( {\frac{1}{2}\left( {x-8} \right),-3y+10} \right)}\). Leave positive \(y\)’s the same. Every point on the graph is shifted left  \(b\)  units. This lesson discusses some of the basic characteristics of linear, quadratic, square root, absolute value and reciprocal functions. Transformed: \(y=\left| {\sqrt[3]{x}} \right|\). In general, transformations in y-direction are easier than transformations in x-direction, see below. Rotated Function Domain:  \(\left[ {0,\infty } \right)\)    Range:  \(\left( {-\infty ,\infty } \right)\). Also, the last type of function is a rational function that will be discussed in the Rational Functions section. Linear—vertical shift up 5. Note that if we wanted this function in the form \(\displaystyle y=a{{\left( {\left( {x-h} \right)} \right)}^{3}}+k\), we could use the point \(\left( {-7,-6} \right)\) to get \(\displaystyle y=a{{\left( {\left( {x+4} \right)} \right)}^{3}}-5;\,\,\,\,-6=a{{\left( {\left( {-7+4} \right)} \right)}^{3}}-5\), or \(\displaystyle a=\frac{1}{{27}}\). The equation will be in the form \(y=a{{\left( {x+b} \right)}^{3}}+c\), where \(a\) is negative, and it is shifted up \(2\), and to the left \(1\). Worksheet 2.4—Parent Functions & Transformations Show all work on a separate sheet of paper. Note that examples of Finding Inverses with Restricted Domains can be found here. 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. For this function, note that could have also put the negative sign on the outside (thus affecting the \(y\)), and we would have gotten the same graph. IMPORTANT NOTE:  In some books, for \(\displaystyle f\left( x \right)=-3{{\left( {2x+8} \right)}^{2}}+10\), they may NOT have you factor out the 2 on the inside, but just switch the order of the transformation on the \(\boldsymbol{y}\). Most of the time, our end behavior looks something like this:\(\displaystyle \begin{array}{l}x\to -\infty \text{, }\,y\to \,\,?\\x\to \infty \text{, }\,\,\,y\to \,\,?\end{array}\) and we have to fill in the \(y\) part. 10. The parent graph quadratic goes up 1 and over (and back) 1 to get two more points, but with a vertical stretch of 12, we go over (and back) 1 and down 12 from the vertex. On to Absolute Value Transformations – you are ready! The chart below provides some basic parent functions that you should be familiar with. Let’s try to graph this “complicated” equation and I’ll show you how easy it is to do with a t-chart: \(\displaystyle f(x)=-3{{\left( {2x+8} \right)}^{2}}+10\), (Note that for this example, we could move the \({{2}^{2}}\) to the outside to get a vertical stretch of \(3\left( {{{2}^{2}}} \right)=12\), but we can’t do that for many functions.). We can graph various square root and cube root functions by thinking of them as transformations of the parent graphs y=√x and y=∛x. Stretch graph vertically by a scale factor of \(a\) (sometimes called a dilation). Note that this transformation flips around the \(\boldsymbol{y}\)–axis, has a horizontal stretch of 2, moves right by 1, and down by 3. Range: \(\left( {0,\infty } \right)\), \(\displaystyle \left( {-1,\,1} \right),\left( {1,1} \right)\), \(y=\text{int}\left( x \right)=\left\lfloor x \right\rfloor \), Domain:\(\left( {-\infty ,\infty } \right)\)