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In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $.

(a) $ f(x) = -\frac{1}{2} (x - 2)^2 + 1 $

(b) $ g(x) = \left[\frac{1}{2} (x -1) \right]^2 - 3 $

(c) $ h(x) = -\frac{1}{2} (x +1)^2 - 1 $

(d) $ k(x) = [2(x + 1)]^2 +4 $

a) Reflect over the $x$ -axis, vertical shrink, shift up by 1 and right by 2

b) Vertical shrink, shift down by 3 and right by 1

c) Reflect over the $x$ -axis, vertical shrink, shift down by 1 and left by 2

d) Vertical stretch, shift up by 4 and left by 1

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we have the graph of a few different functions, and we need to compare these graphs with a graph of X squared and then determine how each function has changed the graph. So the way that we do this is we look at if the graph has been stretched or shrunk if it's been reflected across any access and if it's been moved horizontally or vertically. So let's look at the grab negative one half times a quantity X minus, two squared plus one. So comparing it to X squared, we first see that we are reflecting it across the X axis. So it's going from con cave up to con cave down, and then we are multiplying it by one half. So we're vertically shrinking it, and we have X minus two, which means we're shifting it to the right by two. And then the whole function is plus one, so we're shifting it vertically. One. Now let's look at one half times X minus one. That quantity squared minus three. So since we have one half in front of X, we are vertically shrinking It, since the function is minus three, were shifting it down by three because we have X minus one. We're shifting it to the right by one. Let's look at negative one half times X plus one squared minus one. So, again, this is reflected over the X axis. Since we're multiplying it by one half, it is a vertical shrink because it is X plus one. It is shifted one to the left because it is minus one. We're shifting at one down. Okay. And the last 12 times X plus one that quantity squared. Plus four. So this is a vertical stretch. And the other ones, we've had vertical shrinks. So here we're stretching. We're multiplying by something greater than one. We have X plus one. So we're moving it to the left one. And we have the function plus four. So we're moving it up. Four units