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Question

Solution Steps

1. Understand the question

   10 points - fill in each blank, table, & graph the following rational function:
   $y = \\frac{x+5}{x+2}$
   (1 pt) vertical asymptote: _____________
   (1 pt) horizontal asymptote: _____________
   (1 pt) x-intercept: _____________
   (1 pt) y-intercept: _____________
   (1 pt) hole: _____________
   (1 pt) domain: _____________
   (1 pt) range: _____________
   (1 pt) table
   x | y
   (2 pts) graph grid

2. Explanation

   Step1: Find vertical asymptote

   Set denominator to 0: $x+2=0 \implies x=-2$

   Step2: Find horizontal asymptote

   Degrees equal, use leading coeffs: $\frac{1}{1}=1 \implies y=1$

   Step3: Find x-intercept

   Set numerator to 0: $x+5=0 \implies x=-5$

   Step4: Find y-intercept

   Set $x=0$: $y=\frac{0+5}{0+2}=\frac{5}{2}$

   Step5: Check for holes

   No common factors, so no hole.

   Step6: Find domain

   Exclude vertical asymptote: $x
   eq -2$

   Step7: Find range

   Exclude horizontal asymptote: $y
   eq 1$

   Step8: Create value table

   Choose x-values around asymptote:
   | $x$ | $y=\frac{x+5}{x+2}$ |
   |-----|----------------------|
   | $-7$ | $\frac{-7+5}{-7+2}=\frac{-2}{-5}=0.4$ |
   | $-6$ | $\frac{-6+5}{-6+2}=\frac{-1}{-4}=0.25$ |
   | $-5$ | $\frac{-5+5}{-5+2}=0$ |
   | $-3$ | $\frac{-3+5}{-3+2}=\frac{2}{-1}=-2$ |
   | $-1$ | $\frac{-1+5}{-1+2}=\frac{4}{1}=4$ |
   | $0$ | $\frac{0+5}{0+2}=2.5$ |
   | $3$ | $\frac{3+5}{3+2}=\frac{8}{5}=1.6$ |
   | $8$ | $\frac{8+5}{8+2}=\frac{13}{10}=1.3$ |

3. Final answer

   Vertical asymptote: $\boldsymbol{x=-2}$
   Horizontal asymptote: $\boldsymbol{y=1}$
   x-intercept: $\boldsymbol{(-5, 0)}$
   y-intercept: $\boldsymbol{(0, \frac{5}{2})}$
   hole: $\boldsymbol{\text{None}}$
   domain: $\boldsymbol{\{x \in \mathbb{R} \mid x
   eq -2\}}$
   range: $\boldsymbol{\{y \in \mathbb{R} \mid y
   eq 1\}}$

   Table:

   | $x$ | $y$ |
   |-----|-----|
   | $-7$ | $0.4$ |
   | $-6$ | $0.25$ |
   | $-5$ | $0$ |
   | $-3$ | $-2$ |
   | $-1$ | $4$ |
   | $0$ | $2.5$ |
   | $3$ | $1.6$ |
   | $8$ | $1.3$ |

   Graph Instructions:

   1. Draw dashed lines for $x=-2$ (vertical asymptote) and $y=1$ (horizontal asymptote).
   2. Plot the points from the table.
   3. Draw two curves:

   • Left curve: Passes through $(-7,0.4), (-6,0.25), (-5,0), (-3,-2)$, approaching $x=-2$ (downward) and $y=1$ (leftward).
   • Right curve: Passes through $(-1,4), (0,2.5), (3,1.6), (8,1.3)$, approaching $x=-2$ (upward) and $y=1$ (rightward).

Answer

Question Analysis

OCR Text

10 points - fill in each blank, table, & graph the following rational function:
$y = \\frac{x+5}{x+2}$
(1 pt) vertical asymptote: _____________
(1 pt) horizontal asymptote: _____________
(1 pt) x-intercept: _____________
(1 pt) y-intercept: _____________
(1 pt) hole: _____________
(1 pt) domain: _____________
(1 pt) range: _____________
(1 pt) table
x | y
(2 pts) graph grid

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