Part A
<h3>Answer: Translate f(x) 8 units left, or translate f(x) 16 units up</h3>
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Explanation:
Note the point (5,0) is the x intercept of f(x) while (-3,0) is the x intercept of g(x). To move from (5,0) to (-3,0), we move 8 units to the left. Every other point on f(x) will move in the same fashion to ensure f(x) retains its shape, so that it lines up perfectly with g(x).
For the vertical movement, look at a point like (0,-10) which is the y intercept of f(x). If we move (0,-10) up 16 units up, then it moves to (0,6) which is the y intercept of g(x). Subtract the y coordinates and apply absolute value if you do not have access to a graph, or if you want an alternative approach.
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Part B
<h3>Answer: k = 16 for the vertical translation; k = -8 for the horizontal translation</h3>
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Explanation:
The graph of f(x) goes through (0,-10) and (5,0). The equation of this function is f(x) = 2x-10
The equation for g(x) is g(x) = 2x+6
The question is: what is the value of k when g(x) = f(x)+k, which is a general representation of a vertical shift. This is because y = f(x). Plug in the given f and g functions. You should find the following
g(x) = f(x)+k
2x+6 = 2x-10+k
6 = -10+k
6+10 = k
16 = k
k = 16
This matches up with the phrasing "shift f(x) 16 units up" found back in part A.
For the horizontal translation, we would say
g(x) = f(x-k)
so,
f(x) = 2x-10
f(x-k) = 2(x-k)-10
g(x) = f(x-k)
2x+6 = 2(x-k)-10
2x+6 = 2x-2k-10
2x+6 = 2x-10-2k
6 = -10-2k
6+10 = -2k
16 = -2k
-2k = 16
k = 16/(-2)
k = -8
The negative k value tells us to shift in the negative x direction, ie shift to the left. So this matches with the phrasing "shift f(x) 8 units to the left"
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Part C
<h3>Answer: g(x) = f(x)+16 for the vertical translation; g(x) = f(x+8) for the horizontal translation</h3>
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Explanation:
The vertical translation would look like
g(x) = f(x)+k
g(x) = f(x)+16
This is more or less repeating what part B talked about, which was f(x) being translated 16 units upward.
Similarly,
g(x) = f(x-k)
g(x) = f(x-(-8))
g(x) = f(x+8)
represents the horizontal translation of 8 units to the left. Again this is just a repeat.
