7.
$y = \frac{2x^2 + x - 15}{x^2 + 4x + 3}$
vertical asymptote(s) _______________
horizontal or slant asymptote ____________
intercepts _________________________
graph:
coordinate grid with x from -5 to 10 and y from -5 to 10
8. summarize your strategy for graphing rational functions with a step - by - step process.

Question

Accepted Answer

### For Question 7:

#### Vertical Asymptote(s)
# Explanation:
## Step1: Factor numerator and denominator
Factor $2x^2 + x - 15$: We need two numbers that multiply to $2\times(-15)=-30$ and add to $1$. Those numbers are $6$ and $-5$. So, $2x^2 + x - 15 = 2x^2 + 6x - 5x - 15 = 2x(x + 3) - 5(x + 3) = (2x - 5)(x + 3)$.
Factor $x^2 + 4x + 3$: We need two numbers that multiply to $3$ and add to $4$. Those numbers are $1$ and $3$. So, $x^2 + 4x + 3 = (x + 1)(x + 3)$.
The function becomes $y=\frac{(2x - 5)(x + 3)}{(x + 1)(x + 3)}$. Cancel the common factor $(x + 3)$ (note $x
eq - 3$ to avoid division by zero). So the simplified function is $y=\frac{2x - 5}{x + 1}$, with a hole at $x=-3$.
## Step2: Find vertical asymptotes
Vertical asymptotes occur where the denominator of the simplified function is zero (and the numerator is not zero there). Set $x + 1 = 0$, so $x=-1$.
# Answer: $x = - 1$

#### Horizontal or Slant Asymptote
# Explanation:
## Step1: Compare degrees of numerator and denominator
The degree of the numerator (after canceling common factors) is $1$ (from $2x - 5$) and the degree of the denominator (from $x + 1$) is also $1$. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
## Step2: Calculate the ratio of leading coefficients
The leading coefficient of the numerator is $2$ (from $2x$) and the leading coefficient of the denominator is $1$ (from $x$). So the horizontal asymptote is $y=\frac{2}{1}=2$.
# Answer: $y = 2$

#### Intercepts
##### x - intercept:
# Explanation:
## Step1: Set $y = 0$ and solve for $x$
Set $\frac{2x - 5}{x + 1}=0$. A fraction is zero when the numerator is zero (and the denominator is not zero). So set $2x - 5 = 0$.
## Step2: Solve for $x$
$2x - 5 = 0\Rightarrow 2x=5\Rightarrow x=\frac{5}{2}=2.5$. Check the denominator at $x = \frac{5}{2}$: $\frac{5}{2}+1=\frac{7}{2}
eq0$, so it's valid.
# Answer: $x=\frac{5}{2}$ (or $(\frac{5}{2},0)$)

##### y - intercept:
# Explanation:
## Step1: Set $x = 0$ and solve for $y$
Substitute $x = 0$ into the simplified function $y=\frac{2(0)-5}{0 + 1}=\frac{-5}{1}=-5$.
# Answer: $y=-5$ (or $(0, - 5)$)

### For Question 8:
# Explanation:
1. **Factor Numerator and Denominator**: Factor both the numerator and the denominator of the rational function. This helps identify common factors (which create holes) and simplifies the function for further analysis.
2. **Find Holes**: A hole occurs at $x = a$ if $(x - a)$ is a common factor of the numerator and denominator. To find the $y$-coordinate of the hole, substitute $x = a$ into the simplified function (after canceling the common factor).
3. **Find Vertical Asymptotes**: After canceling common factors, vertical asymptotes occur where the denominator of the simplified function is zero (and the numerator is not zero there). Set the denominator equal to zero and solve for $x$.
4. **Find Horizontal/Slant Asymptotes**:
   - If the degree of the numerator ($n$) is less than the degree of the denominator ($m$): Horizontal asymptote is $y = 0$.
   - If $n=m$: Horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
   - If $n=m + 1$: There is a slant (oblique) asymptote, found by dividing the numerator by the denominator (using polynomial long division) and ignoring the remainder.
5. **Find Intercepts**:
   - **x - intercept**: Set $y = 0$ and solve for $x$ (the numerator of the simplified function is zero, and the denominator is not zero at that $x$).
   - **y - intercept**: Set $x = 0$ and solve for $y$ (substitute $x = 0$ into the simplified function).
6. **Plot Key Points**: Plot the intercepts, asymptotes, and any holes. Then, choose test points in the intervals defined by the vertical asymptotes and holes to determine the behavior of the graph (whether it's above or below the horizontal asymptote, etc.).
7. **Sketch the Graph**: Using the asymptotes, intercepts, holes, and test point behavior, sketch the graph of the rational function.

# Answer:
1. Factor numerator and denominator.
2. Identify holes (from common factors).
3. Find vertical asymptotes (denominator of simplified function = 0).
4. Determine horizontal/slant asymptotes (using degrees of numerator/denominator).
5. Find x - intercept (set $y = 0$) and y - intercept (set $x = 0$).
6. Plot key points (intercepts, asymptotes, holes) and test intervals for graph behavior.
7. Sketch the graph using the above information.

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QUESTION IMAGE

Question

Explanation:

For Question 7:

Vertical Asymptote(s)

Step1: Factor numerator and denominator

Step2: Find vertical asymptotes

Step1: Compare degrees of numerator and denominator

Step2: Calculate the ratio of leading coefficients

Step1: Set \(y = 0\) and solve for \(x\)

Step2: Solve for \(x\)

Answer:

Horizontal or Slant Asymptote