Analyzing Quadratic Functions
describe how the graphs of the following functions relate to the graph of y=x^2.
Could someone please tell me what that means? I sort of understand it, but I want to get it perfectly straight.
y = (x+5)^2
Since it's in the brackets, that would mean go five units horizontally. And would it go negative because doesn't it go in the opposite direction?:S
y = x^2-7. Is -7 the slope?
y = 5+(x-2)^2. This one is confusing.
You are correct on the first one. The new function is x^2 if we use a value 5 less than originally. That is, the graph is shifted 5 units to the left.
y = x^2 - 7
is just the graph of y=x^2, but shifted down 7 units.
Now you come upon shifting in both directions.
shift 2 units right, then 5 units up.
In general, the graph of y = f(x) is shifted h units to the right and k units up if
(y-k) = f(x-h)
You can see that the new function is just f(x) if we adjust the values of x and y by h and k, respectively.
If you want to play around with graphs, go to
rechneronline dot de slash function-graphs
You can type in a function f(x) of your choosing (they start out showing x^2). Then, just substitute (x-2) for x and see how the graph shifts.
It's also cool because you can show both graphs at once, in different colors.
To understand how the graphs of the given functions relate to the graph of y = x^2, let's break it down one by one:
1. y = (x+5)^2
In this function, the "+5" inside the parentheses represents a horizontal shift of the graph to the left by 5 units. So, instead of the vertex of the parabola being at (0, 0) like in y = x^2, it will now be at (-5, 0). The negative sign in front of the 5 does not indicate a negative direction but rather a shift to the left. The shape and concavity of the graph will remain the same as y = x^2, just shifted to the left.
2. y = x^2 - 7
In this function, the "-7" at the end of the equation represents a vertical shift of the graph downward by 7 units. So, instead of the vertex of the parabola being at (0, 0) like in y = x^2, it will now be at (0, -7). The value (-7) does not indicate the slope, but rather the change in the y-coordinate. Again, the shape and concavity of the graph will remain the same as y = x^2, just shifted downward.
3. y = 5 + (x-2)^2
In this function, the "+5" at the beginning of the equation represents a vertical shift of the graph upward by 5 units. So, instead of the vertex of the parabola being at (0, 0) like in y = x^2, it will now be at (0, 5). The expression "(x-2)^2" inside the parentheses denotes a horizontal shift of the graph to the right by 2 units. So, the vertex of the parabola will now be at (2, 5). The order of operations is important here: first, the graph is shifted to the right by 2 units, and then it is shifted upward by 5 units.
Remember, the shape and concavity of the graph, which are essential features of quadratic functions, remain the same as y = x^2 in all these cases. It's the shifting up, down, left, or right that changes its position.