Graph the rational function $y=\frac{x}{2 x+2}$. Which quadrant does neither branch of the rational function pass through?
A. Quadrant 1
B. Quadrant 4
C. Quadrant 3
D. Quadrant 2
Answer (2)
The rational function y = 2 x + 2 x passes through Quadrants 1, 2, and 3, but does not pass through Quadrant 4. Therefore, the answer is Quadrant 4.
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Find the vertical asymptote: x = − 1 .
Find the horizontal asymptote: y = 2 1 .
Analyze the sign of y in the intervals x < − 1 , − 1 < x < 0 , and 0"> x > 0 .
Determine that the function passes through Quadrants 1, 2, and 3, and does not pass through Quadrant 4. The answer is Q u a d r an t 4 .
Explanation
Understanding the Problem The given rational function is y = 2 x + 2 x . We need to determine which quadrant the graph of the function does NOT pass through. The quadrants are defined as follows:
Quadrant 1: 0, y>0"> x > 0 , y > 0
Quadrant 2: 0"> x < 0 , y > 0
Quadrant 3: x < 0 , y < 0
Quadrant 4: 0, y<0"> x > 0 , y < 0
Finding the Vertical Asymptote First, let's find the vertical asymptote by setting the denominator equal to zero:
2 x + 2 = 0
2 x = − 2
x = − 1
So, the vertical asymptote is x = − 1 .
Finding the Horizontal Asymptote Next, let's find the horizontal asymptote by considering the limit of the function as x approaches infinity:
lim x → ∞ 2 x + 2 x = lim x → ∞ x ( 2 + x 2 ) x = lim x → ∞ 2 + x 2 1 = 2 + 0 1 = 2 1
So, the horizontal asymptote is y = 2 1 .
Finding the X-Intercept Now, let's find the x-intercept by setting y = 0 and solving for x :
0 = 2 x + 2 x x = 0
So, the x-intercept is x = 0 .
Finding the Y-Intercept The y-intercept is found by setting x = 0 :
y = 2 ( 0 ) + 2 0 = 2 0 = 0
So, the y-intercept is y = 0 .
Analyzing the Sign of y Let's analyze the sign of y in different intervals determined by the vertical asymptote and x-intercept:
Interval 1: x < − 1 . Let's test x = − 2 : 0"> y = 2 ( − 2 ) + 2 − 2 = − 4 + 2 − 2 = − 2 − 2 = 1 > 0 . So, in this interval, 0"> y > 0 . This means the function passes through Quadrant 2.
Interval 2: − 1 < x < 0 . Let's test x = − 0.5 : y = 2 ( − 0.5 ) + 2 − 0.5 = − 1 + 2 − 0.5 = 1 − 0.5 = − 0.5 < 0 . So, in this interval, y < 0 . This means the function passes through Quadrant 3.
Interval 3: 0"> x > 0 . Let's test x = 1 : 0"> y = 2 ( 1 ) + 2 1 = 2 + 2 1 = 4 1 > 0 . So, in this interval, 0"> y > 0 . This means the function passes through Quadrant 1.
Identifying the Missing Quadrant Based on the sign analysis, the function passes through Quadrants 1, 2, and 3. Therefore, the quadrant that the function does not pass through is Quadrant 4.
Final Answer The rational function y = 2 x + 2 x does not pass through Quadrant 4.
Examples
Rational functions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can model the relationship between distance, rate, and time, or in electrical engineering, they can describe the impedance of a circuit. Understanding the behavior of rational functions, including their asymptotes and intercepts, helps in analyzing and predicting the behavior of these real-world systems. By analyzing the quadrants that the function passes through, we can understand the sign of the function in different intervals, which can be useful in optimization problems or stability analysis.
