Answer:
(a) Graph 1 and Graph 3
(b) Graph 1, Graph 2 and Graph 3
(c) Graph 2 and Graph 4
Step-by-step explanation:
Vertical Asymptotes
- Equation: x = a
- A vertical asymptote occurs at the x-value(s) that make the denominator of a rational function zero.
Horizontal Asymptotes
- Equation: y = b
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there is a slant asymptote).
- If the degree of the numerator is equal to the degree of the denominator, the asymptote is the result of dividing the highest degree term of the numerator by the highest degree term of the denominator.
Slant Asymptotes
- Equation: y = mx + c
- A slant asymptote occurs when the degree of the numerator polynomial is exactly one greater than the degree of the denominator polynomial.
- A function with a slant asymptote can never have a horizontal asymptote.
- The equation of the slant asymptote is the quotient of the division of the numerator of the function by its denominator.
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Graph (i)
[tex]y=\dfrac{x^3+x}{x^2-3}[/tex]
Vertical asymptotes:
[tex]\implies x^2-3=0[/tex]
[tex]\implies x^2=3[/tex]
[tex]\implies x=\pm \sqrt{3}[/tex]
No horizontal asymptote as the degree of the numerator is greater than the degree of the denominator.
The function has a slant asymptote as the degree of the numerator polynomial is exactly one greater than the degree of the denominator polynomial.
Graph (ii)
[tex]y=\dfrac{4x}{x^2+1}[/tex]
No vertical asymptote as the denominator can never equal zero.
Horizontal asymptote at y = 0 as the degree of the numerator is less than the degree of the denominator.
No slant asymptote as the degree of the numerator polynomial is not exactly one greater than the degree of the denominator polynomial.
Graph (iii)
[tex]y=\dfrac{x^2}{x^2-9}[/tex]
Vertical asymptotes:
[tex]\implies x^2-9=0[/tex]
[tex]\implies x^2=9[/tex]
[tex]\implies x=\pm 3[/tex]
As the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = 1
No slant asymptote as the degree of the numerator polynomial is not exactly one greater than the degree of the denominator polynomial.
Graph (iv)
[tex]y=\dfrac{x^4}{x^2+4}[/tex]
No vertical asymptote as the denominator can never equal zero.
No horizontal asymptote as the degree of the numerator is greater than the degree of the denominator.
No slant asymptote as the degree of the numerator polynomial is not exactly one greater than the degree of the denominator polynomial.
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Part (a)
The graphs of every rational function has a vertical asymptote.
- Graph 1 and Graph 3 have vertical asymptotes.
Part (b)
The graph of every rational function has at least one vertical, horizontal, or slant asymptote.
- Graph 1, Graph 2 and Graph 3 have at least one asymptote.
Part (c)
The graph of a rational function can have at most one vertical asymptote.
- Graph 2 and Graph 4 do not have any vertical asymptotes.
