WJEC Core 1 Maths - Key Facts

Question	Answer
Formula for the gradient of a line joining two points	$m=\frac{y_2-y_1}{x_2-x_1}$
The quadratic equation formula for solving $ax^2+bx+c=0$	$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
The midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ is...	$\left ( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right )$ Think of this as the mean of the coordinates $(x_1, y_1)$ and $(x_2, y_2)$
A line has gradient $m$ . A line perpendicular to this will have a gradient of...	$\frac{-1}{m}$
If we know the gradient of a line and a point on the line, a formula to work out the equation of the line is...	$y-y_1=m(x-x_1)$
Formula for the distance between two points...	$\sqrt{\left ( x_2-x_1 \right )^2 + \left ( y_2-y_1 \right )^2}$
To find where two graphs intersect each other...	... solve their equations simultaneously.
In a right-angled triangle, $\cos{\theta}=...$	$\frac{adjacent}{hypotenuse}$
In a right-angled triangle, $\sin{\theta}=...$	$\frac{opposite}{hypotenuse}$
In a right-angled triangle, $\tan{\theta}=...$	$\frac{opposite}{adjacent}$
To simplify $\frac{a}{\sqrt{b}}$ ... (a.k.a. 'rationalising the denominator')	Multiply by $\frac{\sqrt{b}}{\sqrt{b}}$
To simplify $\frac{a}{b+\sqrt{c}}$ ... (a.k.a. 'rationalising the denominator')	Multiply by $\frac{b-\sqrt{c}}{b-\sqrt{c}}$
$\left(\sqrt{m} \right)^{3}=...$	$\left(\sqrt{m} \right)^{3}=\sqrt{m}\sqrt{m}\sqrt{m}=m\sqrt{m}$
$\sqrt{a}\times \sqrt{b}=...$	$\sqrt{a}\times \sqrt{b}=\sqrt{ab}$
$\frac{\sqrt{a}}{\sqrt{b}}=...$	$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$
To find the gradient of a curve at any point, use...	Differentiation
Parallel lines have the same...	Gradient
To find the gradient of the line $ax+by+c=0$ ...	Rearrange into the form $y=mx+c$ . The value of $m$ is the gradient.
Where is the vertex of the graph $y=\left ( x+a \right )^2+b$ ?	$\left ( -a,b \right )$
The discriminant of $ax^2+bx+c$ is...	$b^2-4ac$
The discriminant of a quadratic equation tells us...	How many roots (or solutions) it has. This will be how many times it crosses the $x$ -axis
If a quadratic equation has two distinct real roots, what do we know about the discriminant?	$b^2-4ac>0$
If a quadratic equation has two equal roots, what do we know about the discriminant?	$b^2-4ac=0$
If a quadratic equation has no real roots, what do we know about the discriminant?	$b^2-4ac<0$
Here is the graph of $y=x^2-8x+7$ . Use it to solve the quadratic inequality $x^2-8x+7>0$ Image: 130ce81d-812c-4c26-adf3-3249e2f6daae.gif (image/gif)	$x<1$ or $x>7$ These are the red sections of the curve. Note - do not write $x<1$ and $x>7$ - the word 'and' implies $x$ would need to be $<1$ and $>7$ at the same time... which is clearly not possible!
The formula for differentiating by first principles...	$\frac{dy}{dx} = \lim_{\delta x\rightarrow 0}\left (\frac{f\left(x+\delta x\right)-f(x)}{\delta x} \right )$
If $y=ax^n$ , then $\frac{dy}{dx} =...$	$\frac{dy}{dx} =anx^{n-1}$
If $\left (x+a \right )$ is a factor of $f(x)$ , then...	$f(-a)=0$ This is known as the Factor Theorem
If the remainder, when $f(x)$ is divided by $(x+a)$ is R, then...	$f(-a)=R$ This is known as the Remainder Theorem
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]
How would you use the second derivative, $\frac{d^2 y}{dx^2}$ to determine the nature of the stationary points on a graph?	Substitute the $x$ co-ordinates of the stationary points into $\frac{d^2 y}{dx^2}$ . If you get a positive answer, it's a MIN. If you get a negative answer, it's a MAX.
A function is said to be 'increasing' when its gradient is...	Positive
A function is said to be 'decreasing' when its gradient is...	Negative
$a^0=?$	$a^0=1$
$a^m \times a^n = ?$	$a^m \times a^n = a^{m+n}$
$a^m \div a^n = ?$	$a^m \div a^n = a^{m-n}$
$\left( a^m \right) ^n=?$	$\left( a^m \right) ^n=a^{mn}$
$a^{-n}=?$	$a^{-n}=\frac{1}{a^n}$
$a^{\frac{m}{n}}=?$	$a^{\frac{m}{n}}=\left (\sqrt[n]{a} \right )^m$
What does $n!$ mean?	$n!=n(n-1)(n-2)\times \ldots \times 3 \times 2 \times 1$ For example, $4!=4\times 3\times 2\times 1=24$

Next up

WJEC Core 1 Maths - Key Facts

Description

Resource summary

0 comments

Similar

	Created by Daniel Cox over 9 years ago