Question 8 of 10 2 Points Which of the following rational functions is graphed below? 1 A.
4 B.
1 C.
1 D.
The Correct Answer and Explanation is:
The correct answer is:
D. F(x)=1(x−4)2F(x) = \frac{1}{(x – 4)^2}F(x)=(x−4)21
Explanation
To determine the correct rational function that corresponds to the given graph, we need to analyze the key features of the graph shown:
1. Vertical Asymptote:
The graph has a vertical asymptote at x=4x = 4x=4. This occurs in rational functions when the denominator becomes zero (division by zero). So, we are looking for a function whose denominator is zero when x=4x = 4x=4. This rules out:
- Option A: 1(x+4)2\frac{1}{(x + 4)^2}(x+4)21 → vertical asymptote at x=−4x = -4x=−4
- Option B: 4×2\frac{4}{x^2}x24 → vertical asymptote at x=0x = 0x=0
- Option C: 14×2\frac{1}{4x^2}4×21 → vertical asymptote at x=0x = 0x=0
Only Option D, 1(x−4)2\frac{1}{(x – 4)^2}(x−4)21, has a vertical asymptote at x=4x = 4x=4.
2. Shape and Behavior:
The graph is symmetric around the vertical asymptote, and the function values are always positive (approaching infinity as x→4x \to 4x→4 from both sides and decreasing to zero as xxx moves away from 4). This is characteristic of a squared denominator like (x−4)2(x – 4)^2(x−4)2, which ensures the output is always positive, just like 1×2\frac{1}{x^2}x21.
3. No x-intercepts:
There is no x-intercept (the graph never touches the x-axis), which confirms that the numerator is a non-zero constant (here, 1), and the denominator never causes the output to be zero.
Therefore, based on the location of the vertical asymptote and the overall shape of the graph, the rational function that matches the graph is:F(x)=1(x−4)2\boxed{F(x) = \frac{1}{(x – 4)^2}}F(x)=(x−4)21
