f(x)=1/2x and g(x) = f(x) +2. How does adding 2 to f(x) impact the graph

Question

Translates the graph of f(x) = 1/2x up 2 units
Translates the graph of f(x( = 1/2x down 2 units
It reflects the graph of f(x)=1/2x over the x-axis and translator up 2 units

Answer 1

To analyze the impact of the function \( g(x) = f(x) + 2 \) where \( f(x) = \frac{1}{2}x \), let’s break it down.

1. Original function: The function \( f(x) = \frac{1}{2}x \) is a linear function with a slope of \( \frac{1}{2} \), which means it rises slowly. The graph of \( f(x) \) passes through the origin (0,0).

2. Transformation: When you create \( g(x) = f(x) + 2 \), you are adding 2 to the output of the function \( f(x) \). This means for every input \( x \), the output of \( g(x) \) will be equal to the output of \( f(x) \) plus 2.

3. Effect on the graph: Adding a constant value to a function translates (or shifts) the graph vertically. Specifically, adding 2 to \( f(x) \) translates the graph of \( f(x) \) up by 2 units.

So, the correct interpretation is:

Translates the graph of \( f(x) = \frac{1}{2}x \) up 2 units.

In conclusion, \( g(x) = f(x) + 2 \) shifts the graph of \( f(x) \) upward by 2 units.