Shifting Graphs

The accompan...

Question

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.

Accepted Answer

Welcome back everyone to another video. Write the new equation after shifting the graph of Y equals x squared is shown in the figure. In this problem, we're given a parabola. It is given to us in blue. It is y equals x 2. And what we notice is that we are shifting it to form a parabola in red. And specifically what we notice is that we're given the new vertex of a parabola, we're not performing. Any stretching or compressing we're simply performing a horizontal shift and then the vertical shift right to get this new coordinate of the vertex. So now let's equal to shifts. First of all, the horizontal one. The starting parabola is, let's call it F of X equals X2. Whenever we are performing a horizontal shift, we're simply evaluating F of X plus A, and we have to recall two cases. Now, if A is positive, this is a shift to the left. If A is negative, this is a shift to the right by the absolute value of a units. So now what do we notice? Well, the original parabola given to us in blue has coordinates of 0.0. X is 0 and Y is 0. Now, the new X value is -25, right? So essentially we can say that our X. Plus a in this case is going to be X, and then we have to understand that if we're performing the shift to the left, then A must be positive, right? So instead of taking -25, we're going to take the absolute value of -25, which gives us 25. So our first step is to evaluate. F of X + 25, which represents the shift to the left by 25 units to go from 0 to -25. So F of X plus 25 is simply obtained by replacing X with X + 25. So we get X plus 25 squared. Now this new form represents a parabola that has a vertex at -25.0. So at this point we are only performing the horizontal shift, right? 25 units to the left because we have plus 25. Additionally, we want to shift our parabola up by 12 units because the Y coordinate is 12 instead of the original 0. So now for the vertical shift. We have to recall that we're simply evaluating the original function F of X. And adding a constant C. Now if C is positive, we're shifting it up. By C units, if C is negative, we're shifting it down. By the absolute value of C units. So, in this case, we're shifting it up by 12 units because we're going from 0 to 12, meaning we want to evaluate. The previous function. F of X plus 25, and we want to add our constant C, which in this case is 12, right? And that will give us the final form which is going to be X +. 25 squared and then finally we're adding 12 to take into account the vertical shift and this is our new equation, right? So the new equation is going to be Y equals X plus 25 2 + 12. So the horizontal shift 25 units to the left and the vertical shift 12 units upward. Let's label our final answer and thank you for watching.

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.

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Key Concepts

Graph Shifting

Quadratic Functions

Vertex Form of a Quadratic

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