Notice how we must input the value \(x=2\) to get the output value \(y=0\) the \(x\)-values must be 2 units larger because of the shift to the right by 2 units. The graph is the basic quadratic function shifted 2 units to the right, so The vertex used to be at \((0,0)\), but now the vertex is at \((2,0)\). functions that have periods longer or shorter than 2, we can do so by multiplying the independent variable t by a value w ( called angular frequency ). Notice that the graph is identical in shape to the \(f(x)=x^2\) function, but the \(x\)-values are shifted to the right 2 units. The graph of the absolute value function for real numbers. Over and get you to G, which is exactly what we already got.\): Graph of a parabola There are some functions that do not have reflection symmetry across the y axis. 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 you take x is equal to negative two, the absolute value of that is going to be two. If we were to unflip G, it would look like this. Instructor This right over here is the graph of y is equal to absolute value of x which you might be familiar with. 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. The function g g shifts the basic graph down 3 3 units and the function h h shifts the basic graph up 3 3 units. 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. Now plot the points and compare the graphs of the functions g g and h h to the basic graph of f(x) x2 f ( x) x 2, which is shown using a dashed grey curve below. Visually on a graph, this is represented as a flip over the x-axis, moving each point on the original curve to the. It's just exactly what F of X is, but flipped over the X axis. The reflection of the function f(x) sqrt(x) over the x-axis is achieved by changing the sign of each y-value (f(x)) in the original function, resulting in undefined, 0, -1, -2 for the x-values negative 1, 0, 1, 4 respectively. 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 Figure 3-9: Vertical and horizontal reflections of a function. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. G can be thought of as a stretched or compressed version of F of X is equal to Another transformation that can be applied to a function is a reflection over the x or y-axis.
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