In this video, we will be using function shift rules to graph some quadratic

functions. Here we have y=2 times quantity (x-4) squared, minus 1, y=negative of

(x+3) quantity squared, plus 2, and y=-4 times (x-4) quantity squared, plus 8. And here’s our first function. y=2 times (x-4) quantity squared minus 1. What this

will be is a shift of y=2x-squared. We have taken the function y=2x-squared, and we

have replaced x with (x-4). This shifts it to the right 4. We have subtracted 1 from the formula. This shifts it down 1. And y=2x-squared, because of the 2, is a

vertical stretch by a factor of 2. And here is y=2x-squared. We have a vertex at

(0,0), and we have a point at x=1, y=2, and a point at x=-1, y=2. And now we shift these 3 points 4 to the right, and down 1. We start with the vertex, the lowest

point of the parabola. We go to the right 4, and down 1. Which gives us a vertex of

(4,-1). And the other two points are shifted 4 to the right and down 1 as well. Or, remember we have the shape of y=2x-squared, so we go over 1 from the new vertex, up

2 to get another point, over left 1 from the vertex and up 2 to get another point.

We have our shifted graph, y=2 times the quantity (x-4) squared minus 1. Now let’s look at our second graph. y=negative of (x+3) quantity squared, plus 2. That

negative in front of the quantity is the same as -1 times the quantity. So what we

have here is a shift of y=-1x squared. And -1x squared is just an upside down parabola, and we are shifting that upside down parabola left 3, because we have

replaced x with (x+3), and up 2 because we have added 2 to the formula. And I have

y=-x squared graphed here. It has a vertex at (0,0) and when we go 1 unit to the right, we go down 1 unit, 1 unit to the left and down 1 unit. Remember that when we

take the negative of a function, it reflects it across the x-axis. So y=-x-squared,

is y=x squared reflected down across the x-axis. Now we shift that upside-down parabola 3 to the left and up 2. The vertex goes 1, 2, 3 to the left, up 2, to the

new location of (-3,2). And then we have the two other points. We go 1 unit to the

right, down 1, 1 unit to the left, down 1, to replicate the shape of y=-x squared. And there’s our graph! Our last graph is y=-4 times the quantity (x-4) squared,

plus 8. This will be similar to the last graph, except instead of shifting -x-

squared, we are shifting y=-4x squared. And -4x squared is much like -x squared except it is vertically stretched by a factor of 4. Since we have replaced x with

(x-4), we are shifting that upside down stretched parabola right 4, and since we

have added 8, we are shifting it up 8. So here I have y=-4x squared graphed. It’s an upside-down parabola, it has a vertex of (0,0). When we go 1 to the right we go

down 4. When we go 1 to the left we go down 4. Now I am going to shift it. And

here’s our graph. The vertex is shifted right 4 and up 8. So it has a location of (4,8) up here. And we have two points. Go 1 right, down 4. And go 1 to the left and

down 4. And we get that same shape, shifted up to the new location.