Graph each function. See Examples 1 and 2. g(x)=(1/2)x^2

Question

Accepted Answer

Hello, everybody doing. All right. Today, today we're going to bring on this question that states for the following function sketch its graph where our function is H of X is equal to open parentheses, one divided by five closed parentheses, multiplying X squared. Now, the first thing I wanna do here is identify if there is any parent function. And what I mean by that is if is there any way that the current function we have can be written in an even simpler way, a different type of function that our current function is based on. And in this case, I'll leave our parent function F of X. And for our case, our parent function is going to be X squared. So it's the same as our function with the one divided by five removed because the one divided by five is just a transformation of the X X squared graph. So I'll draw a quick general shape of what our parent function looks like just so we know that this is the graph that we will be basing our H FX graph on. We know that the A X squared graph or the F of X graph is a parabola with an origin at or a vertex at the origin zero comma zero. Now let's shift focus back to H O X H of X is equal to open parentheses, one divided by five closed parentheses X squared. Now this means that the F of X function is shrunk vertically by five units, but we have the same vertex. So this is why we want to identify the parent function because we know what the graph, the general shape of our graph is going to look like. And then we know that all we did was transform it with the one divided by five in front. That transformation is just shrinking it vertically by five units. So now we'll do A T chart and this is to test different points. So I'll test different X values to see what the points in our graph would look like. And to do that, I do a T chart where I have an X column on the left and A Y column on the right. And the X values all tests are negative 15, negative 10, negative 505, 10 and 15. Now for an X of negative 15, all we do is plug it into our H of X function. So instead of having H of H of X, we would have H of negative 15. And then you plug that into the X squared. So you have negative 15 squared divided by five, which would be 45 as the Y and then you do that for every X in our chart. So you plug in the negative 10 and you get a Y of 20 you plug in negative five to get a Y of five and you plug in a zero to get a zero. Now because we have a Parabola, we know that this will be the same on the right side of the Y axis and on the left side of the Y axis. So we know that the Y values are going to repeat. So for a positive five X, we'll have a five. Once again, for a positive 10, we'll have a 20 for a positive 15, we'll have a 45. So now that we have all this information, we can go ahead and start graphing. So we draw a Y axis and an X axis and I put the X axis very close to the end of my Y axis because I know that there won't be any negative Y values here. So I'll draw a dash on the X axis where I believe five, 10 and 15 should be. And I'll do the same thing for the negative side. So I'll draw a dashboard negative five, negative 10 and negative 15 should be. Then as for the Y axis, I'll draw up points or dashes where I believe five should be where I believe 20 should be and where I believe 45 should be. And then I can start plugging in points. So I know we're going to have a point of the origin. That's where our vertex is going to be. We'll have a, a point at five comma five, which is the same on the other side, but at negative five comma five. So every point is going to be mirrored across the Y axis, we have a point at 10 comma 20 which once we mirror mirror it across the axis, it'll be at negative 10 comma 20. And we have a point at 15 comma which once again, once, once we marry it, it will be a negative 15 comma 45. So once we do that, we can start connecting all the points. Now remember our parabola is curved, we just have to look at, we look at our parent function and we see that this is curved around every point. So we try to curve it as much as we can try to get that perfect curve and extend infinitely upwards. And we do the same thing for both sides. So we try to connect every point by curbing it and it's not going to look perfectly because we're doing this by hand. But the general shape is that this will be a parabola with a vertex at the origin. And just like that, we've obtained the graph of our H O X function. It will have the same general shape as the parent function X squared that has been shrunk vertically vertically by five units. So I hope that was helpful. And until next time.

Graph each function. See Examples 1 and 2. g(x)=(1/2)x^2

Verified Solution

Key Concepts

Quadratic Functions

Graphing Techniques

Vertex and Axis of Symmetry