The graph of f(x) = - x ^ 2 is given below. Write the equation of a function g(x) which would shift the graph of f(x) left 3 units and down 1 unit.

Answer 1

To shift the graph of \( f(x) = -x^2 \) left by 3 units and down by 1 unit, we follow these steps:

1. Left Shift: To shift a function \( f(x) \) left by \( h \) units, we replace \( x \) with \( x + h \). In this case, since we want to shift left by 3 units, we replace \( x \) with \( x + 3 \): \[ f(x) = -x^2 \quad \text{becomes} \quad f(x + 3) = -(x + 3)^2. \]

2. Down Shift: To shift a function down by \( k \) units, we subtract \( k \) from the function. Since we want to shift down by 1 unit, we subtract 1: \[ f(x + 3) - 1 = - (x + 3)^2 - 1. \]

Combining these transformations, the function \( g(x) \) that represents the graph shifted left by 3 units and down by 1 unit is: \[ g(x) = - (x + 3)^2 - 1. \]

Thus, the final equation for the function \( g(x) \) is: \[ g(x) = - (x + 3)^2 - 1. \]