![]() Over and get you to G, which is exactly what we already got. You were getting before, you now get the opposite value, and that would flip it We could write this as Y is equal to four times F of X, or you could say Y is equal to four times the absolute value of X, and then we have a negative sign. ![]() If were to unflip G, so this thing right over here, this thing looks like four times F of X. If we were to unflip G, it would look like this. We even flip it over, if we were to unflip G, it would look like this. "Hey, let's first stretch "or compress F." And say, alright, before And you could have done it the other way. So we could say that G of X is equal to, it's not negative absolute value of X, negative four times theĪbsolute value of X. Go from the green to G, you have to multiply this Times the negative value, so it's going even more negative, so what you can see, to When X is equal to negative one, my green function gives me negative one, but G gives me negative four. For a given X, at least for X equals one, G is giving me somethingįour times the value that my green function is giving. It to be the same as G, we want it to be equal to negative four. On this green function, when X is equal to one, the function itself isĮqual to negative one, but we want it, if we want We appropriately stretch or squeeze this green function? So let's think about what's happening. So this is getting usĬloser to our definition of G of X. Whatever the absolute value of X would have gotten you before, you're now going to get the negative of the opposite of it. I'll call this, Y is equal to the negative absolute value of X. So this graph right over here, this would be the graph. So it's just flipped over the X axis, so all the values for any given X, whatever Y you used to get, you're not getting the negative of that. Another common way that the graphs of trigonometric functions are altered is by stretching the graphs. It's just exactly what F of X is, but flipped over the X axis. Let's actually, let's flip it first, so let's say that we have a function that looks like this. We could first try to flip F of X, and then try to stretch or compress it, or we could stretch or compress it first, and then try to flip it. So like always, pause this video and see if you can up yourself with the equation for G of X. Stressed or compressed, but it also is flipped over the X axis. What is the equation for G of X? So you can see F of X is equal to the absolute value of X here in blue, and then G of X, not only does it look ![]() Graph y = x3and do one transformation at a time.G can be thought of as a stretched or compressed version of F of X is equal to Y y 8 8 4 4 x x -4 4 4 Step 3: -4 Step 4: Example: Multiple Transformations Example:Graph using the graph ofy = x3. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 2. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by c. If 0 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by c. x –4 4 Vertical Stretching and Shrinking Vertical Stretching and Shrinking If c > 1 then the graph of y= cf(x) is the graph of y = f(x) stretched vertically by c. Y 4 is the graphof y = x2shrunk vertically by. Then shiftthe graphthree units to the left. Y y 4 4 x x 4 4 – 4 -4 Example: Reflections Example:Graph y =–(x + 3)2using the graph of y = x2. y =f(–x) y = f(x) y =–f(x) The graph of the function y =–f(x) is the graph ofy = f(x)reflected in the x-axis. The graph of the function y = f(–x)is the graphof y = f(x) reflected in the y-axis. y x The graph of a function may be a reflection of the graph of a basic function. ![]() First make a vertical shift 4 units downward. Y y 4 4 x x -4 -4 (–1, –2) Example: Vertical and Horizontal Shifts Example:Graph the function using the graph of. Y 4 4 -4 x Example: Horizontal Shifts Example:Use the graph of f(x) = x3to graphg(x) = (x – 2)3 and h(x) = (x + 4)3. c +c If c is a positive real number, then the graph of f(x + c) is the graph of y = f(x)shifted to the leftc units. Horizontal Shifts y x Horizontal Shifts If c is a positive real number, then the graph of f(x – c) is the graph of y = f(x)shifted to the rightc units. Y 8 4 x 4 -4 -4 Example: Vertical Shifts Example:Use the graph of f(x) = |x| to graph thefunctions g(x) = |x| +3 and h(x) = |x| –4. If c is a positive real number, the graph of f(x) –cis the graph of y = f(x)shifted downwardc units. Vertical Shifts y x VerticalShifts If c is a positive real number, the graph of f(x) + cis the graph of y = f(x)shifted upwardc units. ![]() y = x2 + 3 y = x2 The graph of y = –x2is the reflection of the graph of y = x2 in the x-axis. 8 4 x 4 -4 -4 -8 The graphs of many functions are transformations of the graphs of very basic functions. E N D - Presentation TranscriptĢ.5 Shifting, Reflecting, and Stretching GraphsĮxample: Shift, Reflection y Example: The graph of y = x2+ 3is the graph of y = x2shifted upward three units. ![]()
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