Formula for the gradient of a line joining two points

\[ m=\frac{y_2y_1}{x_2x_1}\]

The quadratic equation formula for solving \[ax^2+bx+c=0\]

\[x=\frac{b\pm \sqrt{b^24ac}}{2a}\]

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...

\[ yy_1=m(xx_1)\]

Formula for the distance between two points...

\[ \sqrt{\left ( x_2x_1 \right )^2 + \left ( y_2y_1 \right )^2} \]

To find where two graphs intersect each other...

... solve their equations simultaneously.

In a rightangled triangle,
\[ \cos{\theta}=... \]

\[\frac{adjacent}{hypotenuse}\]

In a rightangled triangle,
\[ \sin{\theta}=... \]

\[\frac{opposite}{hypotenuse}\]

In a rightangled 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...

\[b^24ac\]

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?

\[b^24ac>0\]

If a quadratic equation has two equal roots, what do we know about the discriminant?

\[b^24ac=0\]

If a quadratic equation has no real roots, what do we know about the discriminant?

\[b^24ac<0\]

Here is the graph of \(y=x^28x+7\).
Use it to solve the quadratic inequality \(x^28x+7>0\)
130ce81d812c4c26adf33249e2f6daae.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 \(\frac{dy}{dx} =...\)

\[\frac{dy}{dx} =anx^{n1}\]

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 \(x\) coordinates 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

\[a^0=?\]

\[a^0=1\]

\[a^m \times a^n = ?\]

\[a^m \times a^n = a^{m+n}\]

\[a^m \div a^n = ?\]

\[a^m \div a^n = a^{mn}\]

\[\left( a^m \right) ^n=?\]

\[\left( a^m \right) ^n=a^{mn}\]

\[a^{n}=?\]

\[a^{n}=\frac{1}{a^n}\]

\[a^{\frac{m}{n}}=?\]

\[a^{\frac{m}{n}}=\left (\sqrt[n]{a} \right )^m\]

What does \(n!\) mean?

\[n!=n(n1)(n2)\times \ldots \times 3 \times 2 \times 1\]
For example, \(4!=4\times 3\times 2\times 1=24\)
