In this video, we will graph four transformations of a given function given the graph of the function. In the first example, we have f of x is shown in red. We want to draw the graph of g of x equals f of x plus three. And notice how I've already determined the three key points on the given function f of x. Because g of x equals f of x plus three and the function values for f are the y-coordinates of these points, in order to graph g of x, We need to add three to each of the y coordinates.
By adding three to the y coordinates, the graph will be shifted up three units. So if we add three to the y coordinate of one, one plus three is four, the corresponding point is negative three comma four, which is this point here. Adding three to this y coordinate, two plus three is five, the corresponding point is negative one comma five. And adding three to this y coordinate, negative two plus three is positive one.
The corresponding point is three comma one on g of x. So this is the graph of g of x, which we can see is f of x shifted up three units. Next, we have the same graph of f of x, but now we want to graph g of x equals the opposite of f of x. Again, remember f of x would be the y values or the y coordinates of these points. And since g of x equals the opposite of f of x, to graph g of x, we need to change the sign of the y coordinates.
By changing the sign of the y coordinates, the graph will be reflected across the x-axis. So for the ordered pair negative three comma one, if we change the sign of the y coordinate, the y coordinate becomes negative one. the corresponding point is negative three comma negative one this point here. Changing the sign of the y coordinate for negative one comma two, the corresponding point is negative one comma negative two this point.
And changing the y coordinate of three comma negative two, the corresponding point is three comma positive two this point here. So this blue graph is the graph of g of x when g of x equals the opposite of f of x. And notice how g of x is a reflection of f of x across the x-axis.
Next we have g of x equals f of the quantity x plus 3. We need to be careful about this transformation. It may look like we should add 3 to each x coordinate of f of x to graph g of x, but that's not the case. Let's say we wanted the function values of f of x and g of x to be the same. Well when x is 3, the function value for f of x would be f of three. So if we wanted g of x to have the same function value of f of three, because g of x equals f of the quantity x plus three, notice how the input for g would have to be zero.
Notice g of zero is equal to f of the quantity zero plus three which does give us f of three. And since zero is three less than three, this shows In order to graph g of x, we have to subtract three from each of the x coordinates of f of x. By subtracting three from the x coordinates, the graph is shifted left three units. So starting on the leftmost point, the x coordinate here is negative three.
Negative three minus three is negative six. The corresponding point on g of x is negative six comma one, this point. Subtracting three from negative one, Negative one minus three is negative four.
The corresponding point is negative four comma two, this point, and subtracting three from this x-coordinate, three minus three is zero. The corresponding point for g of x is zero comma negative two. So this is the graph of g of x, which we can see is f of x shifted left three units. Because this is probably the trickiest transformation, let's look at some notes. If we focus on g of x equals f of the quantity x plus c, if c is less than zero or negative, meaning the form would be f of the quantity x minus c, the graph is shifted right c units.
So if we have subtraction here, the graph is shifted right c units, and if c is greater than zero or positive, like in our case, the form of the function is f of the quantity x plus c, the graph is shifted left c units. So we have addition, the graph is shifted left c units. And for our last example, we have g of x equals f of negative x. To graph g of x, we need to change the sign of the inputs or x values of f of x. If we change the sign of the x coordinates, we have a reflection across the y-axis.
So looking at the leftmost point, if we change the sign of the x coordinate of negative 3, it becomes positive 3. The corresponding point is positive 3 comma 1, which is this point here. Changing the sign of the x coordinate of negative one, we get positive one. The corresponding point is positive one comma two, which is here. And finally, if we change the sign of the x coordinate of positive three, it becomes negative three. The corresponding point is negative three comma negative two, which is this point here.
This is the graph of g of x when g of x is equal to f of negative x. And notice how g of x is a reflection of f of x across the y-axis. I hope you found this helpful.