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Find Vertical and Horizontal Asymptotes of the Rational Function $y = \\frac{3 - 2x}{2 - 3x}$
Mathematics
Grade 10 (Junior High School)
Question Content
What are the vertical and horizontal asymptotes of the graph of $y = \\frac{3 - 2x}{2 - 3x}$?
Correct Answer
B. $x = \\frac{2}{3}$ and $y = \\frac{2}{3}$
Detailed Solution Steps
1
Step 1: Find the vertical asymptote. A vertical asymptote occurs where the denominator of the rational function equals zero (as long as the numerator does not also equal zero at that point). Set the denominator $2 - 3x = 0$.
2
Step 2: Solve for $x$ in $2 - 3x = 0$. Rearrange the equation: $-3x = -2$, so $x = \\frac{2}{3}$. Check the numerator at $x=\\frac{2}{3}$: $3 - 2*(\\frac{2}{3}) = 3 - \\frac{4}{3} = \\frac{5}{3} \\neq 0$, so $x=\\frac{2}{3}$ is the vertical asymptote.
3
Step 3: Find the horizontal asymptote. For a rational function $y = \\frac{ax + b}{cx + d}$ (where $a$ and $c$ are non-zero), the horizontal asymptote is $y = \\frac{a}{c}$. Here, the coefficient of $x$ in the numerator is $-2$, and the coefficient of $x$ in the denominator is $-3$.
4
Step 4: Calculate the horizontal asymptote: $y = \\frac{-2}{-3} = \\frac{2}{3}$.
Knowledge Points Involved
1
Vertical Asymptotes of Rational Functions
A vertical asymptote is a vertical line $x=k$ that the graph of a function approaches but never touches. For rational functions, it exists where the denominator is zero and the numerator is not zero at that $x$-value; this is because the function is undefined at that point, and the output approaches positive or negative infinity as $x$ approaches $k$.
2
Horizontal Asymptotes of Linear Rational Functions
For a rational function in the form $y=\\frac{ax+b}{cx+d}$ (where $a$ and $c$ are non-zero constants), the horizontal asymptote is the line $y=\\frac{a}{c}$. This comes from the fact that as $x$ approaches positive or negative infinity, the constant terms $b$ and $d$ become negligible compared to the linear terms, so the function behaves like $y=\\frac{ax}{cx}$.
3
Solving Linear Equations
A linear equation in one variable is an equation of the form $mx + n = 0$. To solve it, isolate the variable by performing inverse operations (subtracting/adding constants, dividing/multiplying by non-zero coefficients) to find the value of the variable that makes the equation true. This is used to find vertical asymptotes by setting the denominator equal to zero.
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