Let \(f(x)=\dfrac{1}{(x+3)^2}-2\). On the axes below, sketch the graph of \(y=f(x)\), labelling all asymptotes with their equations and axis intercepts with their coordinates. (4 marks) --- 4 WORK AREA LINES (style=lined) --- \(\text{Find asymptotes:}\) \((x+3) \neq 0\ \ \Rightarrow\ \ \text{Asymptote at}\ x=-3\) \(\text{As}\ x \rightarrow \infty, \ \dfrac{1}{(x+3)^2} \rightarrow 0\ \ \Rightarrow \ \ \text{Asymptote at}\ y=-2\) \(y\text{-intercept:}\ x=0\) \(y=\dfrac{1}{(0+3)^2}-2=-\dfrac{17}{9}\) \(x\text{-intercepts:}\ y=0\)
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\(\dfrac{1}{(x+3)^2}-2\)
\(=0\)
\((x+3)^2\)
\(=\dfrac{1}{2}\)
\(x+3\)
\(=\pm\dfrac{1}{\sqrt{2}}\)
\(x\)
\(=-3\pm\dfrac{1}{\sqrt{2}}\)
Functions, 2ADV F2 EQ-Bank 2
A graph of the hyperbola \(y=\dfrac{1}{x+p}+q\) is shown, where \(p\) and \(q\) are constants. Find the values of \(p\) and \(q\) and hence the graphs \(x\)-axis intercept. (2 marks) --- 6 WORK AREA LINES (style=lined) ---
Functions, 2ADV F2 2022 SPEC2 3 MC
The graph of `y=\frac{x^2+2x+c}{x^2-4}`, where `c \in R`, will always have
- two vertical asymptotes and one horizontal asymptote.
- a vertical asymptote with equation `x=-2` and one horizontal asymptote with equation `y=1`.
- one horizontal asymptote with equation `y=1` and only one vertical asymptote with equation `x=2`.
- a horizontal asymptote with equation `y=1` and at least one vertical asymptote.
Functions, 2ADV F2 SM-Bank 9 MC
The graph of the function `f(x)=(3x+2)/(5-x)`, has asymptotes at
- `x=-5,y=(3)/(2)`
- `x=(2)/(3),y=-3`
- `x=5,y=3`
- `x=5,y=-3`
Functions, 2ADV F2 2021 HSC 19
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`
Functions, 2ADV F2 SM-Bank 10 MC
The graph of the function `f(x) = (x - 3)/(2 - x)` has asymptotes
- `x = 3,\ \ \ \ \ \ \ \ \ y = 2`
- `x = -2,\ \ \ \ \ y = 1`
- `x = 2,\ \ \ \ \ \ \ \ \ y = -1`
- `x = 2,\ \ \ \ \ \ \ \ \ y = 1`
Functions, 2ADV’ F2 2007 HSC 3b
- Find the vertical and horizontal asymptotes of the hyperbola
`qquad y = (x − 2)/(x − 4)` and hence sketch the graph of `y = (x − 2)/(x − 4)`. (3 marks)
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- Hence, or otherwise, find the values of `x` for which `(x − 2)/(x − 4) ≤ 3`. (2 marks)
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Show Worked Solution
i. `y = (x − 2)/(x − 4)`
`text(Vertical asymptote at)\ \ x = 4`
| `lim_(x → ∞) (x − 2)/(x − 4)` | `= lim_(x → ∞) (1 − 2/x)/(1 − 4/x)=1` |
`ytext(–intercept)\ = 1/2`
`xtext(–intercept)\ = 2`
ii. `text(Find)\ \ x\ \ text(so that)\ \ (x − 2)/(x − 4) ≤ 3`
| `(x − 2)/(x − 4)` | `= 3` |
| `x − 2` | `= 3x − 12` |
| `2x` | `= 10` |
| `x` | `= 5` |
`=>(5, 3)\ \ text(is the intersection of)\ \ y = 3\ and\ y = (x − 2)/(x − 4)`
`:. (x − 2)/(x − 4) ≤ 3\ \ text(when)\ \ x < 4\ \ text(and)\ \ x ≥ 5.`
Functions, 2ADV’ F2 2015 HSC 5 MC
What are the asymptotes of `y = (3x)/((x + 1)(x + 2))`
| A. | `y = 0,` | `x = −1,` | `x = −2` |
| B. | `y = 0,` | `x = 1,` | `x = 2` |
| C. | `y = 3,` | `x = −1,` | `x = −2` |
| D. | `y = 3,` | `x = 1,` | `x = 2` |