Question

F(x) = x^2. What is g(x)

Answer 1

g(x) = - x² - 4 ⇒ A

Explanation:

Let us revise the reflection and translation of a function

• If the function f(x) reflected across the x-axis, then its image is g(x) = - f(x)

• If the function f(x) reflected across the y-axis, then its image is g(x) = f(-x)

• If the function f(x) translated horizontally to the right by h units, then its image is g(x) = f(x - h)

• If the function f(x) translated horizontally to the left by h units, then its image is g(x) = f(x + h)

• If the function f(x) translated vertically up by k units, then its image is g(x) = f(x) + k

• If the function f(x) translated vertically down by k units, then its image is g(x) = f(x) – k

f(x) = x² is the blue curve

g(x) is its image is the red curve

∵ g(x) is the image of f(x)

∵ f(x) is opened upward

∵ g(x) is opened downward

→ That means the sign of y-coordinates of all points on the blue

graph are opposite

∴ f(x) is reflected about the x-axis

∴ Its image is - f(x)

∵ The vertex of f(x) is (0, 0)

∵ The vertex of g(x) = (0, -4)

→ That means the function translated 4 units down

∴ - f(x) is translated 4 units down

∴ Its image is - f(x) - 4

∴ g(x) = - f(x) - 4

∵ f(x) = x²

∴ g(x) = - x² - 4