The answers are:
Part A:
The 1st type of transformation of the Vertical shift of k: g(x) = f (x) + k
The 2nd type of transformation of the Horizontal shift of k : g (x) = f (x-k)
Part B:
For the vertical translation we have: 16 = k
For the horizontal translation we have: -8 = k
Part C: g(x) = f(x) + 16 or g(x) = f(x - (-8))
What is the linear functions about?
A linear function is one that is often written:
y = a*x + b
Note:
a = the slope
b = the y-intercept.
Since the line passes cross the points (x₁, y₁) and (x₂, y₂):
So the the slope will be:
a = (y₂ - y₁)/(x₂ - x₁)
The line f(x) cross points (5, 0) and (10, 10)
So the slope of f(x) is:
a = (10 - 0)/(10 - 5)
= 10/5 = 2
Note that f(x) = 2*x + b
Since f(10) = 10
10 = 2 x 10 + b
10 = 20 + b
10 - 20 = b = -10
The equation of the line is f(x) = 2*x - 10
Note that g(x) we know that it passes via (-3, 0) and (2, 10)
So the slope is:
a = (10 - 0)/(2 - (-3))
= 10/5
= 2
Since g(x) = 2*x + b and that g(2) = 10 as the line passes via point (2, 10) )
So 10 = 2 x 2 + b
10 = 4 + b
10 - 4 = b
6 = b
So: g(x) = 2*x + 6
Therefore, the equations are:
f(x) = 2x+10
g(x) = 2x + 6
g(x) = 2x + b
Part B:
In vertical translation:
g(x) = f(x) + k
2x + 6 = 2x - 10 + k
6 = -10 + k
6 + 10 = k
16 = k
For horizontal translation:
g(x) = f(x - k)
2x + 6 = 2(x - k) - 10
2x + 6 = 2x - 2*k - 10
6 = -2k - 10
6 + 10 = -2k
16 = -2k
16/-2 = k
-8 = k
Therefore, the answers are:
Part A:
The 1st type of transformation of the Vertical shift of k: g(x) = f (x) + k
The 2nd type of transformation of the Horizontal shift of k : g (x) = f (x-k)
Part B:
For the vertical translation we have: 16 = k
For the horizontal translation we have: -8 = k
Part C: g(x) = f(x) + 16 or g(x) = f(x - (-8))
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