Question

What's the End Behavior, Vertical Asymptote, and Shifts for f(x)=log4 (x+10)−2?

A) End Behavior: Approaches -∞, Vertical Asymptote: x = -10, Shifts: Left 10, Down 2
B) End Behavior: Approaches -∞, Vertical Asymptote: x = 10, Shifts: Left 10, Down 2
C) End Behavior: Approaches ∞, Vertical Asymptote: x = -10, Shifts: Right 10, Down 2
D) End Behavior: Approaches ∞, Vertical Asymptote: x = 10, Shifts: Right 10, Down 2

Answer 1

For the function f(x) = log4(x + 10) - 2, the end behavior is that f(x) approaches infinity as x approaches infinity, the vertical asymptote is x = -10, and the function is shifted left by 10 units and down by 2 units.

Step-by-step explanation:

The student is asking about the end behavior, vertical asymptote, and shifts for the logarithmic function f(x) = log4(x + 10) - 2. To determine these components, we consider the properties of logarithmic functions.

End Behavior: As x approaches infinity, the logarithm function continues to increase, but at a decreasing rate. Therefore, log4(x+10) will also approach infinity as x becomes very large. Subtracting 2 does not change this; it only lowers the graph. Thus, the end behavior is that f(x) approaches infinity as x approaches infinity.

Vertical Asymptote: A vertical asymptote of the function occurs where the input to the logarithm is zero because the log of zero is undefined. So for log4(x+10), this happens when x+10 equals zero, which is when x is -10. Therefore, the vertical asymptote is x = -10.

Shifts: The x+10 inside the logarithm indicates a horizontal shift to the left by 10 units. The -2 outside the logarithm indicates a vertical shift downwards by 2 units.

Combining all these facts, the correct answer is: End Behavior: Approaches infinity, Vertical Asymptote: x = -10, Shifts: Left 10, Down 2, which is option C.