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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Show Worked Solution
\(\dfrac{1}{(x+3)^2}-2\)
\(=0\)
\((x+3)^2\)
\(=\dfrac{1}{2}\)
\(x+3\)
\(=\pm\dfrac{1}{\sqrt{2}}\)
\(x\)
\(=-3\pm\dfrac{1}{\sqrt{2}}\)
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 2023 MET1 3
- Sketch the graph of \(f(x)=2-\dfrac{3}{x-1}\) on the axes below, labelling all asymptotes with their equation and axial intercepts with their coordinates. (3 marks)
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- Find the values of \(x\) for which \(f(x)\leq1\). (1 mark)
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a.
b. \(1<x\leq4\ \ \ \text{or}\ \ \big(1,4\big]\)
Show Worked Solution
a. \(\text{Vertical asymptote when}\ \ x=1\)
\(y\text{-int:}\ y=2-(-3)\ \ \Rightarrow\ \ y=5\)
\(x\text{-int:}\ 2-\dfrac{3}{x-1}=0\ \ \Rightarrow\ \ x=\dfrac{5}{2} \)
\(\text{As}\ \ x \rightarrow \infty, \ \ y \rightarrow 2^{-}; \ \ x \rightarrow -\infty, \ \ y \rightarrow 2^{+} \)
b. \(\text{From the graph:}\)
\(f(x)=1\ \text{when }x=4\)
\(x>1\ \text{to the right of the vertical asymptote}\)
\(\therefore\ f(x)\leq1\ \text{when}\ \ 1<x\leq4\ \ \ \text{or}\ \ \big(1,4\big]\)
♦♦ Mean mark (b) 38%.
MARKER’S COMMENT: Many students did not use their graph from part (a).
MARKER’S COMMENT: Many students did not use their graph from part (a).
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 SM-Bank 20
- Sketch the graph of `y = 1 - 2/(x - 2)` on the axes below. Label asymptotes with their equations and axis intercepts with their coordinates. (3 marks)
- Find the values of `x` for which `1 - 2/(x - 2) >= 3`. (1 mark)
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a.
b. `x in [1, 2)`
Show Worked Solution
a. `text(Asymptotes:)`
`x=2`
`text(As)\ \ x→ +-oo, \ y→1\ \ =>\ text(Asymptote at)\ \ y=1`
`ytext(-intercept at)\ (0,2)`
`xtext(-intercept at)\ (4,0)`
b. `text(By inspection of the graph:)`
`1-2/(x-2) >=3\ \ text(for)\ \ x in [1, 2)`
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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Show Worked Solution
`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 EQ-Bank 9
Consider the function `f(x) = 1/(4x - 1)`.
- Find the domain of `f(x)`. (1 mark)
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- Sketch `f(x)`, showing all asymptotes and intercepts? (2 marks)
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- `{text(all real)\ x, x!=1/4}`
-
Show Worked Solution
| i. | `4x – 1` | `!= 0` |
| `x` | `!= 1/4` |
`:.\ text(Domain:)\ {text(all real)\ x, x!=1/4}`
ii. `text(When)\ \ x = 0, \ y = −1`
`text(As)\ \ x -> ∞, \ y -> 0^+`
`text(As)\ \ x -> −∞, \ y -> 0^−`
Functions, 2ADV F2 SM-Bank 12
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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Show Worked Solution
| `(2x+1)/(x-1)` | `=(2x-2+3)/(x-1)` | |
| `=(2(x-1)+3)/(x-1)` | ||
| `=2 + 3/(x-1)` |
COMMENT: Manipulation of the equation makes graphing much easier.
`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`
|
Functions, 2ADV F2 EQ-Bank 11
On the axes below, sketch the graph of `f(x) = (2x-2)/(x + 1)`.
Label all axis intercepts. Label each asymptote with its equation. (4 marks)
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Show Worked Solution
| `(2x-2)/(x + 1)` | `=(2(x+1) – 4)/(x+1)` |
| `=2- 4/((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`
Functions, 2ADV F2 SM-Bank 10
Consider the function `f(x) = x/(4 - x^2)`.
- Identify the domain of `f(x)`. (1 mark)
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- Sketch the graph `y = f(x)`, showing all intercepts and asymptotes. (3 marks)
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- `text(Domain:)\ \ text(all)\ x,\ x != +- 2`
-
Show Worked Solution
i. `4-x^2 !=0`
`=> x!= 2 or -2`
`:.\ text(Domain:)\ \ text(all)\ x,\ x != +- 2`
ii. `text(Asymptotes at)\ \ x = +- 2`
`text(Passes through)\ (0,0)`
COMMENT: Note that `x->2^(–)` is a short way of writing as `x` approaches 2 from the negative (or left-hand) side. This notation can save time when required.
| `text(As)` | `\ \ x -> 2^(-),` | `\ y -> oo` |
| `\ \ x -> 2 ^(+),` | `\ y -> – oo` |
| `text(As)` | `\ \ x -> -2^ (-),` | `\ y -> oo` |
| `\ \ x -> -2^ (+),` | `\ y -> -oo` |
| `text(As)` | `\ \ x -> oo,` | `\ y -> 0` |
| `\ \ x -> – oo,` | `\ y -> 0` |
 |
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 2012 HSC 13b
- Find the horizontal asymptote of the graph `y=(2x^2)/(x^2 + 9)`. (1 mark)
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- Without the use of calculus, sketch the graph `y=(2x^2)/(x^2 + 9)`, showing the asymptote found in part (i). (2 marks)
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- `text(Horizontal asymptote at)\ y = 2`
-
Show Worked Solution
| i. | `y` | `= (2x^2)/(x^2 +9)` |
| `= 2/(1 + 9/(x^2))` |
`text(As)\ \ x -> oo,\ y ->2`
`text(As)\ \ x -> – oo,\ y -> 2`
`:.\ text(Horizontal asymptote at)\ y = 2`
| ii. | `text(At)\ \ x = 0,\ y = 0` |
`f(x) = (2x^2)/(x^2 + 9) >= 0\ text(for all)\ x`
`f(–x) = (2(–x)^2)/((–x)^2 + 9) = (2x^2)/(x^2 + 9) = f(x)`
`text(S)text(ince)\ \ f(x) = f(–x) \ \ =>\ text(EVEN function)`
