So let's think about this. And we could do that with a bunch of points. We could see that g of 0, which is right there-- let me do it in a color you can see-- g of 0 is equivalent to f of negative 2. So let me write that down. We could keep doing that. We could say g of 1, which is right over here. This is 1. So I think you see the pattern here. So we could say that g of x is equal to f of-- well it's going to be 2 less than x.
So f of x minus 2. So this is the relationship. And it's important to realize here. When I get f of x minus 2 here-- and remember the function is being evaluated, this is the input.
When I subtract the 2, this is shifting the function to the right, which is a little bit counter-intuitive unless you go through this exercise right over here. If it was f of x plus 2 we would have actually shifted f to the left.
Now let's think about this one. This one seems kind of wacky. So first of all, g of x, it almost looks like a mirror image but it looks like it's been flattened out. So let's think of it this way. Let's take the mirror image of what g of x is. This page will be removed in future.
Stretching and Reflecting Transformations. Preview Assign Practice. Practice Now. Algebra Graphs and Functions. Quick Tips. Found a content error? Tell us. Horizontal and vertical translations, as well as reflections, are called rigid transformations because the shape of the basic graph is left unchanged, or rigid. Functions that are multiplied by a real number other than 1, depending on the real number, appear to be stretched vertically or stretched horizontally.
This type of non-rigid transformation is called a dilation A non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally. The function g is steeper than the basic squaring function and its graph appears to have been stretched vertically. The function h is not as steep as the basic squaring function and appears to have been stretched horizontally. Otherwise, the graph will be stretched vertically.
If the factor a is negative, then it will produce a reflection as well. This results in a reflection and a dilation. Then translate this graph 5 units to the right and 3 units down. In summary, given positive real numbers h and k :. Graph the given function.
Identify the basic function and translations used to sketch the graph. Then state the domain and range. Graph the piecewise functions. Write an equation that represents the function whose graph is given. Match the graph the given function definition. Use the transformations to graph the following functions. What happens to the graph when the denominator of k is very large? To translate a function vertically is to shift the function up or down.
In general, a vertical translation is given by the equation:. The original function we will use is:. To translate a function horizontally is the shift the function left or right. While vertical shifts are caused by adding or subtracting a value outside of the function parameters, horizontal shifts are caused by adding or subtracting a value inside the function parameters. The general equation for a horizontal shift is given by:.
Again, the original function is:. Reflections produce a mirror image of a function. As an example, let the original function be:. Consider an example where the original function is:. The third type of reflection is a reflection across a line. In algebra, equations can undergo scaling, meaning they can be stretched horizontally or vertically along an axis.
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