Question
Question A rational function has the given information. Choose the equation that could represent the same rational function. horizontal asymptote: y=32 vertical asymptote: x=−2 hole at x=5 (1 point) Responses y=3x2−15x2x2−10x y is equal to the fraction with numerator 3 x squared minus 15 x and denominator 2 x squared minus 10 x y=3x2−12x−152x2−6x−20 y is equal to the fraction with numerator 3 x squared minus 12 x minus 15 and denominator 2 x squared minus 6 x minus 20 y=3x4−12x−152x6−6x3−20 y is equal to the fraction with numerator 3 x to the 4th power minus 12 x minus 15 and denominator 2 x to the 6th power minus 6 x cubed minus 20 y=x2−4x−5x2−3x−10
Answers
GPT-4o mini
To find a rational function that meets the specified criteria (horizontal asymptote at \( y = 32 \), vertical asymptote at \( x = -2 \), and a hole at \( x = 5 \)), we need to analyze the general structure of the rational function.
1. **Horizontal Asymptote**: To achieve a horizontal asymptote of \( y = 32 \), the ratio of the leading coefficients of the numerator and denominator must equal \( 32 \). This suggests that if the degrees of the numerator and denominator are the same, the leading coefficient of the numerator should be \( 32 \times \) the leading coefficient of the denominator.
2. **Vertical Asymptote**: A vertical asymptote at \( x = -2 \) implies that the denominator must contain a factor of \( (x + 2) \).
3. **Hole**: A hole at \( x = 5 \) means both the numerator and denominator must have the factor \( (x - 5) \).
### Constructing the Function
Based on these points:
- Let the numerator be:
\[
N(x) = k(x - 5)(\text{other factors})
\]
- Let the denominator be:
\[
D(x) = (x - 5)(x + 2)(\text{other factors})
\]
### Horizontal Asymptote Calculation
To achieve a horizontal asymptote at \( y = 32 \), we'll ensure the leading coefficients balance out. For simplicity, let's denote the additional factors just as \( a \) for the numerator and \( b \) for the denominator's remaining factors.
Thus, we can express a general rational function:
\[
y = \frac{k(x - 5)(a)}{(x - 5)(x + 2)(b)}
\]
Canceling the \( (x - 5) \) gives:
\[
y = \frac{k \cdot a}{(x + 2)(b)}
\]
To ensure the horizontal asymptote is \( 32 \), we would let \( \frac{k \cdot a}{b} = 32 \).
### Choosing the Right Option
Now we need to check which of the provided options fits this structure:
1. **Option 1**: \( y = \frac{3x^2 - 15x}{2x^2 - 10x} \)
- This simplifies to an expression with \( \frac{3(x - 5)}{2(x - 5)(x + 2)}\). There is a hole at \( x=5 \) but does not achieve the horizontal asymptote \( y=32 \).
2. **Option 2**: \( y = \frac{3x^2 - 12x - 15}{2x^2 - 6x - 20} \)
- This does not exhibit a clear behavior for the specified asymptotes and holes.
3. **Option 3**: \( y = \frac{3x^4 - 12x - 15}{2x^6 - 6x^3 - 20} \)
- Degrees are different and will not have the desired horizontal asymptote.
4. **Option 4**: \( y = \frac{x^2 - 4x - 5}{x^2 - 3x - 10} \)
- This does not fit our criteria since terms are not easily simplified.
### Finding the Equation
Let's check each option thoroughly for their characteristics to narrow down to one that includes a hole, vertical asymptote, and horizontal asymptote.
Based on the assessments, the best possible rational function that could match the criteria is:
After analyzing the responses, **Option 1**: \( y = \frac{3x^2 - 15x}{2x^2 - 10x} \) satisfies having a hole at \( x=5 \) and a vertical asymptote \( x=-2 \).
Thus, the answer is:
\[
\boxed{y = \frac{3x^2 - 15x}{2x^2 - 10x}}
\]
1. **Horizontal Asymptote**: To achieve a horizontal asymptote of \( y = 32 \), the ratio of the leading coefficients of the numerator and denominator must equal \( 32 \). This suggests that if the degrees of the numerator and denominator are the same, the leading coefficient of the numerator should be \( 32 \times \) the leading coefficient of the denominator.
2. **Vertical Asymptote**: A vertical asymptote at \( x = -2 \) implies that the denominator must contain a factor of \( (x + 2) \).
3. **Hole**: A hole at \( x = 5 \) means both the numerator and denominator must have the factor \( (x - 5) \).
### Constructing the Function
Based on these points:
- Let the numerator be:
\[
N(x) = k(x - 5)(\text{other factors})
\]
- Let the denominator be:
\[
D(x) = (x - 5)(x + 2)(\text{other factors})
\]
### Horizontal Asymptote Calculation
To achieve a horizontal asymptote at \( y = 32 \), we'll ensure the leading coefficients balance out. For simplicity, let's denote the additional factors just as \( a \) for the numerator and \( b \) for the denominator's remaining factors.
Thus, we can express a general rational function:
\[
y = \frac{k(x - 5)(a)}{(x - 5)(x + 2)(b)}
\]
Canceling the \( (x - 5) \) gives:
\[
y = \frac{k \cdot a}{(x + 2)(b)}
\]
To ensure the horizontal asymptote is \( 32 \), we would let \( \frac{k \cdot a}{b} = 32 \).
### Choosing the Right Option
Now we need to check which of the provided options fits this structure:
1. **Option 1**: \( y = \frac{3x^2 - 15x}{2x^2 - 10x} \)
- This simplifies to an expression with \( \frac{3(x - 5)}{2(x - 5)(x + 2)}\). There is a hole at \( x=5 \) but does not achieve the horizontal asymptote \( y=32 \).
2. **Option 2**: \( y = \frac{3x^2 - 12x - 15}{2x^2 - 6x - 20} \)
- This does not exhibit a clear behavior for the specified asymptotes and holes.
3. **Option 3**: \( y = \frac{3x^4 - 12x - 15}{2x^6 - 6x^3 - 20} \)
- Degrees are different and will not have the desired horizontal asymptote.
4. **Option 4**: \( y = \frac{x^2 - 4x - 5}{x^2 - 3x - 10} \)
- This does not fit our criteria since terms are not easily simplified.
### Finding the Equation
Let's check each option thoroughly for their characteristics to narrow down to one that includes a hole, vertical asymptote, and horizontal asymptote.
Based on the assessments, the best possible rational function that could match the criteria is:
After analyzing the responses, **Option 1**: \( y = \frac{3x^2 - 15x}{2x^2 - 10x} \) satisfies having a hole at \( x=5 \) and a vertical asymptote \( x=-2 \).
Thus, the answer is:
\[
\boxed{y = \frac{3x^2 - 15x}{2x^2 - 10x}}
\]