Question

Let (x) = x^2 Find a formula for a function g whose graph is obtained from the graph of y = f(x) after the following sequence of transformations:

• Shift left 5 units
• Reflection across the y-axis.
• Shift down 2 units
• Vertical scaling by a factor of 4.
• Reflection across the x-axis.
g(x) = ?

Answer 1

The formula for the function g(x), obtained from the graph of y = f(x) after a sequence of transformations, is g(x) = -4(-x + 5)^2 + 8.

Step-by-step explanation:

To find the formula for the function g(x) that represents the graph of y = f(x) after the given transformations, we step-by-step apply each transformation to the original function.

1. To shift left 5 units, we replace x in the function with (x + 5): g(x) = f(x+5) = (x + 5)^2.
2. To reflect across the y-axis, we replace x in the function with (-x): g(x) = f(-x + 5) = (-x + 5)^2.
3. To shift down 2 units, we subtract 2 from the function: g(x) = (-x + 5)^2 - 2.
4. To vertically scale by a factor of 4, we multiply the function by 4: g(x) = 4((-x + 5)^2 - 2) = 4(-x + 5)^2 - 8.
5. To reflect across the x-axis, we multiply the function by -1: g(x) = -4(-x + 5)^2 + 8.

Therefore, the formula for the function g(x) after the given transformations is g(x) = -4(-x + 5)^2 + 8.