<p>pg. 674 #20. It bascially shows a graph and the equation f(x)=(x+h)+k and they want you to find (hk) but i have no idea. any help?</p>
<p>The graph of g(x) is the same as the graph of f(x), but it has been moved 3 units to the left and 2 units up. h represents how far the graph has been shifted to the left or right, while k represents how far the graph has been moved up or down. In this case, h is 3 and k is 2. Therefore, hk=6, or E.</p>
<p>I confused the two graphs. There is a shift to the right and down in forming the graph of g(x). h=-3 and k=-2, so hk=6.</p>
<p>This is probably the hardest function problem I've seen on the SAT, because I forgot the rules concerning function shifts. Not many people know them, and it is still a tricky problem if you get the movements mixed up. If you know how functions move, you can get this question right just by looking at it. Most people however, know the behavior of a function when a constant is added to the end. I will explain the problem in very slow steps with lengthy explanations for those who don't know function movement with f(x+h).</p>
<p>You could work it out by hand if you don't know h or k just by looking at the equations...which I had to do (I had to solve for h).</p>
<p>The first thing you should have noticed is that the graphs are essentially the same; g is basically f is shifted h units to the right and k units down. For f to become g, it has to go two units down and three to the right.</p>
<p>A constant added to the end of a function by itself shifts the graph that number up/down. This is common knowledge. To allign f(x) with g(x), you can set the constant k at the end of g(x) equal to -2. The reason why it's -2 is because g is essentially f(x) shifted two units down, and h units to the right. So now, you should be looking at two identical graphs, 3 units apart.</p>
<p>This is the hard part which took literally 3 minutes. Those who know graph behavior would know that h=-3, and the answer is hk=(-2)(-3)=6 However, most people don't know this...and the only way to find h is to solve for it. </p>
<p>You can solve for h by figuring out the equation of g(x), then solving for h. Keep in mind that we have already solved k.</p>
<p>g(x)=f(x+h)+k = g(x)=f(x+h)-2</p>
<p>We can figure out the equation of g because f is given, f(x)=x^3-4x.</p>
<p>Substitute (x+h) in for every x in the function f, which yields g(x): </p>
<p>g(x)=(x+h)^3-4(x+h)-2 (recall g(x)=f(x+h)-2)</p>
<p>Look at the graph of g. The coordinates (2,1) lie on the graph. These are values to plug in for the above equations. (recall coordinates are defined as (x, g(x)), therefore x=2 and g(x)=1) This is a key step.</p>
<p>1=(2+h)^3-4(2+h)-2</p>
<p>Simplify by distrubting the -4.</p>
<p>1=(2+h)^3-8-4h-2</p>
<p>Add like terms.</p>
<p>1=(2+h)^3-4h-10</p>
<p>Isolate the -10.</p>
<p>11=(2+h)^3-4h</p>
<p>Another key step. You could solve this equation by hand, but you can use your graphing calculator to solve equations too. Because this post is far too long and the vast majority of SAT takers on CC have a graphing calculator, I won't explain how to solve this equation by hand. PM me if you'd like to know how.</p>
<p>Graph Y1 as (2+h)^3-4h, and Y2 as 11.</p>
<p>Find the interesection. You should come up with -3. Thus, h=-3.</p>
<p>The problem asks for kh. Given h=-3 and k=-2, we finally end up with the answer of (-2)(-3)=6.</p>
<p>The moral of the story: Know function behavior. All of this work could have been avoided if one knew how the functions relate. Collegeboard provided equations and coordinates just as a gift to those who were clueless about function behavior. However-it is very easy to mess up the directions, especially if you cannot tell that f(x) MUST become g(x), not the other way around.</p>
<p>cool, thanks alot guys for the help!</p>
<p>thanks for the help...even my math whiz sister was having a trouble with this question..</p>
<p>Isn’t the standard shift equation f(x) = (x-h) + k to determine shifts? I had 6 but then changed it to -6 because I thought that since it was a positive shift (to the right):</p>
<p>-h = 3
h = -3</p>
<p>can anyone tell me what’s wrong with this reasoning? or if a made a silly mistake? (i make lots of those lol…)</p>
<p>^ That would be right, except that the equation the problem gives you is f(x)=(x**+h**)+k. Using your equation you can rewrite that as f(x)=(x-(-h))+k, so they’re looking for the opposite of the h in your equation.</p>