Question

The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x):

Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points)

Part B: Solve for k in each type of transformation. (4 points)

Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points)

Answer 1

A) horizontal translation; vertical translation

B) H: k = -8; V: k = 16

C: H: g(x) = 2(x +8) -10; V: g(x) = (2x -10) +16; both ⇒ g(x) = 2x +6

Explanation:

Part A:

A linear function can be translated to another linear function by translating it horizontally, vertically, or some combination of those.

• translation right k units: g(x) = f(x -k)
• translation up k units: g(x) = f(x) +k
• translation right h and up k units: g(x) = f(x -h) +k

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Part B:

Point (x, y) on the graph of f(x) will be translated to point (x+h, y+k) to become a point on the graph of g(x). 'h' is the amount of translation to the right; 'k' is the amount of translation up.

horizontal translation: (x, y) = (5, 0) on f(x) moves to (x +k, y) = (-3, 0) on g(x).

k = -3 -5 = -8 . . . horizontal translation of f(x)

vertical translation: (x, y) = (0, -10) on f(x) moves to (x, y+k) = (0, 6) on g(x).

k = 6 -(-10) = 16 . . . vertical translation of f(x)

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Part C:

The equation for f(x) has y-intercept -10 and a slope of 2:

f(x) = 2x -10

Horizontal translation using the formula in Part A gives you ...

g(x) = f(x+8) = 2(x +8) -10

Vertical translation gives you ...

g(x) = f(x) +16 = (2x -10) +16

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Additional comment

The translated equations simplify to the same thing in both cases:

g(x) = 2x +6