Question

to produce function g, function f was reflected over the x - axis and dropdown options: shifted to the left 2 units, shifted to the right 2 units, shifted down 2 units, shifted up 2 units. function g is defined as dropdown

Explanation:

Step1: Analyze vertical shift

First, observe the y - intercepts. Function \( f \) and \( g \): the y - intercept of \( f \) and \( g \) shows a vertical change. After reflection over x - axis, the vertical shift: looking at the graph, the function \( g \) is above \( f \) in terms of vertical position. The vertical shift of \( g \) relative to the reflected \( f \) is up 2 units. Also, check the vertical movement: when we reflect over x - axis and then shift up 2 units, the graph of \( g \) matches.

Step2: Confirm the shift

The reflection over x - axis changes the sign of the function. Then, the vertical shift: from the options, "shifted up 2 units" is correct. For the second part (assuming \( f(x) \) is an exponential function, say \( f(x)=a^x + k \), after reflection over x - axis: \( -f(x)=-a^x - k \), then shifting up 2 units: \( g(x)=-f(x)+2 \). If we assume \( f(x) = (\frac{1}{2})^x\) (since it's a decreasing exponential), reflection over x - axis: \( -(\frac{1}{2})^x\), then shift up 2 units: \( g(x)=-(\frac{1}{2})^x + 2 \). But from the graph, the key is the vertical shift. The correct shift is up 2 units.

Answer:

shifted up 2 units