Describe how the graphs of each of the following can be obtained by transforming the graph of $y=f(x)$.$$ It is easy to tell that you can obtain this graph by vertically stretching the graph of $y=x$ by a factor of $a$, and then moving it up by $b$: Gradient-intercept formįrom the previous lesson, we discussed the general form of a linear function is $$ f(x) = ax + b. All the output values change by k k units. Using these operations, we can define two different algebraic forms of a line graph. Given a function f (x) f ( x), a new function g(x) f (x)+k g ( x) f ( x) + k, where k k is a constant, is a vertical shift of the function f (x) f ( x). Right: Vertically reflecting the graph of $y=x+1$. The graph of $-y=f(x)$ is the reflection of $y=f(x)$ about the x-axis.Ĭaption Left: Horizontally reflecting the graph of $y=x+1$.The graph of $y=f(-x)$ is the reflection of $y=f(x)$ about the y-axis.See this in action and understand why it happens. To dilate a graph toward or away from the y-axis, you divide $x$ by a positive factor $a$. We can reflect the graph of yf(x) over the x-axis by graphing y-f(x) and over the y-axis by graphing yf(-x). Just like translations, you can dilate horizontally or vertically. However, this is a challenge for those who always seek reasons! Dilationĭilation means to stretch or contract a graph. You certainly do not have to memorise this. If you find this too confusing, please do not worry! It is a very confusing concept (I took several years to finally understand it). caption Translating a graph by moving the origin. If you want to move your graph up, you instead move the coordinate plane down. Here comes the secret: you are not moving the graph, but you are changing the perspective by moving the coordinate plane itself. Still, if you want to move $y=3x$ to the right by three units, you have to subtract three from $x$! For example, if you want to move a point, say $A(0,1)$, to the right by three units, you have to add three to the x-coordinates (so it becomes $A\rq(3, 1)$). See how this is applied to solve various problems. We can even reflect it about both axes by graphing y-f (-x). Note: there are no method marks awarded for any transformation questions no matter how many marks are being awarded, they are all answer marks.ĭilation from the x-axis by a factor of aĭilation from the y-axis by a factor of aĮxample 3.Some of you might be thinking that this rule is counter-intuitive. We can reflect the graph of any function f about the x-axis by graphing y-f (x) and we can reflect it about the y-axis by graphing yf (-x). In methods we are generally concerned with - reflections in the x-axis, y-axis and line y=x Reflections simply flip the graph relative to a certain line. All dilation factors are positive.Ī dilation from the y-axis by a factor larger than one ( > 1) will result in the graph being stretched away from the y-axis.Ī dilation from the y-axis by a factor less than one ( 1) will result in the graph being stretched away from the x-axis.Ī dilation from the x-axis by a factor less than one ( < 1) will result in the graph being compressed towards the x-axis. Translations are when we shift the entire graph left, right, up or down.ĭilations stretch or compress the graph. There are three types of transformations which we will be dealing with: translations, dilations and reflections.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |