WJEC Core 3 Maths - Key Facts

Question	Answer
$\sec x\equiv ?$	$\sec x\equiv \frac{1}{\cos x}$
$\text{cosec } x\equiv ?$	$\text{cosec }x\equiv \frac{1}{\sin x}$
$\cot x\equiv ?$	$\cot x\equiv \frac{1}{\tan x}$
Write an identity which links together $\sec \theta$ and $\tan \theta$	$\sec^2 \theta \equiv 1 + \tan^2 \theta$
Write an identity which links together $\text{cosec }\theta$ and $\cot \theta$	$\text{cosec}^2 \theta \equiv 1 + \cot^2 \theta$
Sketch the graph of $y=\sin x$ for $0\leq x \leq 360^{\circ}$	Image: 22b29905-169d-49b7-a15e-fb2008405edc (image/png)
Sketch the graph of $y=\cos x$ for $0\leq x \leq 360^{\circ}$	Image: ac8d2b28-b5fb-41bd-b113-467047db9024 (image/png)
Sketch the graph of $y=\tan x$ for $0\leq x \leq 360^{\circ}$	Image: 79a474de-dc94-46df-a11e-5935aac4ef83 (image/png)
Sketch the graph of $y=\sin^{-1}x$ . What is its domain and range?	Domain $[-1,1]$ Range $[-\frac{\pi}{2},\frac{\pi}{2}]$ Image: 46416b02-d2d9-4852-bb37-91477271b128.gif (image/gif)
Sketch the graph of $y=\cos^{-1}x$ . What is its domain and range?	Domain $[-1,1]$ Range $[0,\pi]$ Image: e9ca3abc-5242-447d-ada2-11dd8839dce1.gif (image/gif)
Sketch the graph of $y=\tan^{-1}x$ . What is its domain and range?	Domain $(-\infty,\infty)$ Range $(-\frac{\pi}{2}, \frac{\pi}{2})$ Image: dffbf049-b3d3-4969-a9d8-c3c9faaf501c.gif (image/gif)
What does $\|x\|<a$ mean?	$-a<x<a$ $\text{NOT } x<\pm a$
What does $\|x\|>a$ mean?	$x>a \text{ or } x<-a$ $\text{NOT } x>\pm a$
Give the definition of a function	A function relates each element of a set with exactly one element of another set
How are the graphs of $y=f(x)$ and $y=f^{-1}(x)$ related?	They are reflections of each other in the line $y=x$ Image: 0e7000cc-088b-4a22-a109-d2e9be5272c9 (image/png)
The domain of $f(x)$ equals the range of ...?	The domain of $f(x)$ equals the range of $f^{-1}(x)$
The range of $f(x)$ equals the domain of ...?	The range of $f(x)$ equals the domain of $f^{-1}(x)$
What is meant by the range of a function?	The set of all possible output values of a function
What is meant by the domain of a function?	All the values that could go into a function
State the inverse function of $e^x$	$\ln x$
Sketch the graph of $y=e^x$ , showing any intersections with the axes. State its domain and range.	The $x$ axis is an asymptote. Domain $(-\infty, \infty)$ Range $(0, \infty)$ Image: 696ef065-c1c8-48d7-8916-a69711f24853.PNG (image/PNG)
Sketch the graph of $y=\ln x$ , showing any intersections with the axes. State its domain and range.	The $y$ axis is an asymptote. Domain $(0, \infty)$ Range $(-\infty, \infty)$ Image: 36cef179-b141-4169-8626-107aa708b941.PNG (image/PNG)
State the inverse function of $\ln x$	$e^x$
Differentiate $e^{f(x)}$ with respect to $x$	$f'(x)e^{f(x)}$
Differentiate $\ln{f(x)}$ with respect to $x$	$\frac{f'(x)}{f(x)}$
Differentiate $\sin {\left(f(x)\right)}$ with respect to $x$	$f'(x)\cos{\left(f(x)\right)}$
Differentiate $\cos {\left(f(x)\right)}$ with respect to $x$	$-f'(x)\sin{\left(f(x)\right)}$
Differentiate $\tan {\left(f(x)\right)}$ with respect to $x$	$f'(x)\sec^2{\left(f(x)\right)}$
State the product rule for differentiating $y=uv$ with respect to $x$ , where $u$ and $v$ are functions of $x$	$\frac{\text{d}y}{\text{d}x}=uv'+vu'$
State the quotient rule for differentiating $y=\frac{u}{v}$ with respect to $x$ , where $u$ and $v$ are functions of $x$	$\frac{\text{d}y}{\text{d}x}=\frac{vu'-uv'}{v^2}$
$\frac{1}{\left(\frac{\text{d}x}{\text{d}y}\right)}=?$	$\frac{\text{d}y}{\text{d}x}$
How would you find the first and second derivatives of the parametric equations $x=f(t)$ and $y=g(t)$	$\frac{\text{d}y}{\text{d}x}=\frac{g'(t)}{f'(t)}$ $\frac{\text{d}^2y}{\text{d}x^2}=\frac{\frac{\text{d}}{\text{d}t}\left(\frac{\text{d}y}{\text{d}x}\right)}{\frac{\text{d}x}{\text{d}t}}$
$\int e^x \, dx =?$	$\int e^x \, dx = e^x+c$
$\int \frac{1}{x} \, dx = ?$	$\int \frac{1}{x} \, dx = \ln x +c$
$\int \sin x \, dx = ?$	$\int \sin x \, dx = -\cos x +c$
$\int \cos x \, dx = ?$	$\int \cos x \, dx = \sin x +c$
What effect will the transformation $y=f(x)+a$ have on the graph of $y=f(x)$ ?	Translation $a$ units in the $y$ direction. i.e. the graph will move UP by $a$ units
What effect will the transformation $y=f(x+a)$ have on the graph of $y=f(x)$ ?	Translation $-a$ units in the $x$ direction. i.e. the graph will move LEFT by $a$ units
What effect will the transformation $y=af(x)$ have on the graph of $y=f(x)$ ?	Stretch, scale factor $a$ in the $y$ direction. i.e. the $y$ values will be multiplied by $a$
What effect will the transformation $y=f(ax)$ have on the graph of $y=f(x)$ ?	Stretch, scale factor $\frac{1}{a}$ in the $x$ direction. i.e. the $x$ values will be divided by $a$ [This could also be described as a 'squash', scale factor $a$ in the $x$ direction]
Sketch the graph of $y=\|x\|$	Image: 4c0acddb-9d3a-4750-a276-047735074528.PNG (image/PNG)

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WJEC Core 3 Maths - Key Facts

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