A)Which graph shows the rational function f(x)=x2+3x+2x2−x−2?(1 point)

Question

Responses

The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through negative left parenthesis 4 comma 1 right parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through Left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1. An open circle is on the left branch of the curve near left parenthesis negative 1 comma negative 0.25 right parenthesis.
Image with alt text: The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through negative left parenthesis 4 comma 1 right parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through Left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1. An open circle is on the left branch of the curve near left parenthesis negative 1 comma negative 0.25 right parenthesis.

The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through left parenthesis negative 4 comma 1 left parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1. An open circle is on the left branch of the curve at left parenthesis negative 2 comma 0 right parenthesis.
Image with alt text: The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through left parenthesis negative 4 comma 1 left parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1. An open circle is on the left branch of the curve at left parenthesis negative 2 comma 0 right parenthesis.

The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5 and a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis. The graph has a dashed line through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 6 comma 0 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at the y axis and curves down through left parenthesis negative 2 comma negative 1 right parenthesis; left parenthesis 0 comma negative 2 right parenthesis; and left parenthesis 1 comma negative 4 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 4 right parenthesis; left parenthesis 4 comma 2 right parenthesis; and left parenthesis 6 comma 1 right parenthesis and curves down as it approaches the dashed line at the y axis.
Image with alt text: The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5 and a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis. The graph has a dashed line through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 6 comma 0 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at the y axis and curves down through left parenthesis negative 2 comma negative 1 right parenthesis; left parenthesis 0 comma negative 2 right parenthesis; and left parenthesis 1 comma negative 4 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 4 right parenthesis; left parenthesis 4 comma 2 right parenthesis; and left parenthesis 6 comma 1 right parenthesis and curves down as it approaches the dashed line at the y axis.

The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through negative left parenthesis 4 comma 1 right parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1.
Image with alt text: The coordinate grid has its x axis labeled from negative 4 to 6 and its y axis labeled from negative 5 to 5. The graph has a dashed line through left parenthesis 2 comma 5 right parenthesis and left parenthesis 2 comma negative 5 right parenthesis and a dashed line through negative left parenthesis 4 comma 1 right parenthesis and left parenthesis 6 comma 1 right parenthesis. The graph has two curved branches. The left branch starts below the dashed line at y equals 1 and curves down through left parenthesis negative 2 comma 0 right parenthesis; left parenthesis 0 comma negative 1 right parenthesis; and left parenthesis 1 comma negative 3 right parenthesis; and curves down as it approaches the dashed line at x equals 2. The right branch starts to the right of the dashed line at x equals 2 and curves down through left parenthesis 3 comma 5 right parenthesis; left parenthesis 4 comma 3 right parenthesis; and left parenthesis 6 comma 2 right parenthesis, and curves down as it approaches the dashed line at y equals 1.

pick an answer

Answer 1

To analyze the rational function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \), we should first determine its characteristics, such as vertical and horizontal asymptotes.

1. Factor the numerator and denominator:
   • The numerator: \( x^2 + 3x + 2 = (x+1)(x+2) \)
   • The denominator: \( 2x^2 - x - 2 \) To factor this, we can look for factors of \( 2 \times -2 = -4 \) that sum to -1. These factors are -2 and +2. Thus, the denominator can be factored by using synthetic division or decomposition, resulting in \( 2x^2 - 4x + 3x - 2 = (2x + 3)(x - 1) \).

So, we have: \[ f(x) = \frac{(x+1)(x+2)}{(2x+3)(x-1)} \]

2. Determine vertical asymptotes:

   • Vertical asymptotes occur where the denominator equals zero, i.e., \( 2x + 3 = 0 \) or \( x - 1 = 0 \).
   • Solving these gives:
     • \( x = -\frac{3}{2} \) (vertical asymptote)
     • \( x = 1 \) (vertical asymptote)

3. Determine horizontal asymptotes:

   • The degrees of the numerator and the denominator are the same. The horizontal asymptote is the ratio of the leading coefficients.
   • The leading coefficient of the numerator is 1 and the leading coefficient of the denominator is 2:
   • Thus, the horizontal asymptote is \( y = \frac{1}{2} \).

4. Intercepts:

   • The x-intercepts occur when the numerator is zero, i.e., \( x + 1 = 0 \) or \( x + 2 = 0 \) gives \( x = -1 \) and \( x = -2 \).
   • The y-intercept occurs when \( x = 0 \): \( f(0) = \frac{(0 + 1)(0 + 2)}{(2(0) + 3)(0 - 1)} = \frac{2}{-3} = -\frac{2}{3} \).

Now, we can match these characteristics with the provided graph descriptions.

• Vertical Asymptotes: Lines at \( x = -1.5 \) and \( x = 1 \).

• Horizontal Asymptote: Line at \( y = \frac{1}{2} \).

• X-intercepts: Points at \( (-1, 0) \) and \( (-2, 0) \).

• Y-intercept: Point at \( (0, -\frac{2}{3}) \).

Given these characteristics, the right graph will have the appropriate behavior: the left branch should start below the line \( y = \frac{1}{2} \) and approach \( x = 1 \) from below. The right branch should start above \( y = \frac{1}{2} \) and curve downwards.

The first option matches that description with the horizontal and vertical asymptotes drawn and the curve behavior as described.

Therefore, the answer is the first graph option.