Question

Let f(x)=x3. Find a formula for a function g whose graph is obtained from f by the given sequence of transformations:

A) g(x)=5(x+4)3
B) g(x)=−5(x−4)3
C) g(x)=−5(x+4)3
D) g(x)=5(x−4)3

Answer 1

The function g(x) can be found by analyzing the given transformations separately and understanding how they affect the original function f(x) = x^3. The correct option is B) g(x) = -5(x - 4)^3.

Step-by-step explanation:

The given functions are transformations of the function f(x) = x^3. We need to find a formula for the function g(x) that represents the given transformations in options A, B, C, and D:

A) g(x) = 5(x + 4)^3

B) g(x) = -5(x - 4)^3

C) g(x) = -5(x + 4)^3

D) g(x) = 5(x - 4)^3

To find g(x) for each option, we need to analyze the transformation separately. The number in front of the function represents a vertical stretch or compression, while the expression inside the function represents a horizontal translation.

A) In option A, the function is stretched vertically by a factor of 5. The expression (x + 4) inside the function represents a horizontal translation of 4 units to the left.

B) In option B, the function is stretched vertically by a factor of -5, which also reflects the function across the x-axis. The expression (x - 4) inside the function represents a horizontal translation of 4 units to the right.

C) In option C, the function is stretched vertically by a factor of -5 and reflected across the x-axis. The expression (x + 4) inside the function represents a horizontal translation of 4 units to the left.

D) In option D, the function is stretched vertically by a factor of 5. The expression (x - 4) inside the function represents a horizontal translation of 4 units to the right.

Therefore, based on the given transformations, the correct option is B) g(x) = -5(x - 4)^3.