Aussie Maths & Science Teachers: Save your time with SmarterEd
Sketch the graph of `y=4/(x-3)`. (3 marks)
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`text{Vertical asymptote at}\ \ x=3`
`text{As}\ \ x->oo, \ y->0`
`text{Horizontal asymptote at}\ \ y=0`
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 & \ \ 0\ \ & \ \ 2\ \ & \ \ 4\ \ & \ \ 5\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -1 & -\frac{4}{3} & -4 & 4 & 2\\
\hline
\end{array}
Sketch the graph of `y=2/(3-x)`. (3 marks)
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`text{Vertical asymptote at}\ \ x=3`
`text{As}\ \ x->oo, \ y->0`
`text{Horizontal asymptote at}\ \ y=0`
\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 & \ \ 0\ \ & \ \ 2\ \ & \ \ 4\ \ & \ \ 5\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & \frac{1}{2} & \frac{2}{3} & 2 & -2 & -1\\
\hline
\end{array}
Sketch the graph of `y=3/(x+1)`. (2 marks)
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`text{Vertical asymptote at}\ \ x=-1`
`text{As}\ \ x->oo, \ y->0`
`text{Horizontal asymptote at}\ \ y=0`
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -3 & -2 & -1 & \ \ 0\ \ & \ \ 1\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{3}{2} & -3 & ∞ & 3 & \frac{3}{2} \\
\hline
\end{array}
Sketch the graph of `y=1/(x-2)`. (2 marks)
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`text{Vertical asymptote at}\ \ x=2`
`text{As}\ \ x->oo, \ y->0`
`text{Horizontal asymptote at}\ \ y=0`
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & \ \ 0\ \ & \ \ 1\ \ & \ \ 2\ \ & \ \ 3\ \ & \ \ 4\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{1}{2} & -1 & ∞ & 1 & \frac{1}{2} \\
\hline
\end{array}
Sketch the graph of `f(x) = (2x+1)/(x-1)`. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation. (4 marks)
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| `(2x+1)/(x-1)` | `=(2x-2+3)/(x-1)` | |
| `=(2(x-1)+3)/(x-1)` | ||
| `=2 + 3/(x-1)` |
`text(Asymptotes:)\ \ x = 1,\ \ y = 2`
`text(As)\ \ x->oo,\ \ y->2(+)`
`text(As)\ \ x->-oo,\ \ y->2(-)`
`text(As)\ \ x->-1 (-),\ \ y->-oo`
`text(As)\ \ x->-1 (+),\ \ y->oo`
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i. `text{Vertical asymptote at}\ \ x=0`
`text{Horizontal asymptote at}\ \ y=2`
ii.
i. `y=2-1/x`
`text{Vertical asymptote at}\ \ x=0`
`text{As}\ x->oo, \ 1/x -> 0\ \ => 2-1/x -> 2`
`text{Horizontal asymptote at}\ \ y=2`
ii.
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \ & \ \ 1\ \ & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & \frac{5}{2} & 3 & ∞ & 1 & \frac{3}{2} \\
\hline
\end{array}
Sketch the graph of `y=2/x+2`.
Clearly mark all asymptotes. (3 marks)
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\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \ & \ \ 1\ \ & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 1 & 0 & ∞ & 4 & 3 \\
\hline
\end{array}
Sketch the graph of `y=-2/x`. (2 marks)
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\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \ & \ \ 1\ \ & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 1 & 2 & ∞ & -2 & -1 \\
\hline
\end{array}
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \ & \ \ 1\ \ & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{3}{2} & -3 & ∞ & 3 & \frac{3}{2} \\
\hline
\end{array}
Without using calculus, sketch the graph of `y = 2 + 1/(x + 4)`, showing the asymptotes and the `x` and `y` intercepts. (3 marks)
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`text(Asymptotes:)\ x = -4`
`text(As)\ \ x -> ∞, y -> 2`
`ytext(-intercept occurs when)\ \ x = 0:`
`y = 2.25`
`xtext(-intercept occurs when)\ \ y = 0:`
`2 + 1/(x + 4) = 0 \ => \ x = -4.5`
