Answer:
- The graph that represents a reflection of f(x) across the x-axis is the blue line on the picture attached.
Explanation:
The function f(x) is:
Which is an exponential function with these features:
- y-intercept: f(0) = 6(0.5)⁰ = 6(1) = 6
- multiplicative rate of change: 0.5 (the base of the exponential term), which means that it is a decaying function (decreasing)
- Horizontal asympote: y = 0 (this is the limit of f(x) when x approaches +∞.
The reflection of f(x) across the x-axis is a function g(x) such that g(x) = - f(x).
Thus, the reflection of f(x) across the x-axis is:
The features of that function are:
- Limit when x approaches - ∞: -∞ (thus the function starts in the third quadrant).
- y-intercerpt: g(0) = -6 (0.5)⁰ = -6(1)= - 6.
- Horizontal asympote: y = 0 (this is the limit of f(x) when x approaches +∞.
- Note that the function never touches the x-axis, thus the function increases from -∞, crosses the y-axis at (0, -6) and continous growing approaching the x-axis but never touchs it. So, this is an increasing frunction, that starts at the third quadrant and ends in the fourth quadrant.
With those descriptions, you can sketch the graph, which you can see in the figure attached. There you have the function f(x) (the red increasing line) and its reflection across the x-axis (the blue increasing line).
