Edexcel Core 1 Maths - Key Facts

Daniel Cox
Flashcards by Daniel Cox, updated more than 1 year ago
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Key points and important formulae for the Edexcel GCE Maths C1 module

Resource summary

Question Answer
Formula for the gradient of a line joining two points \[ m=\frac{y_2-y_1}{x_2-x_1}\]
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 \( (x_2, y_2) \)
The quadratic equation formula for solving \[ax^2+bx+c=0\] \[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\]
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.
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 \(x^2-8x+7>0\) 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!
If \(y=ax^n\), then \(\frac{dy}{dx} =...\) \[\frac{dy}{dx} =anx^{n-1}\]
If \(y=ax^n\), then \(\int y\; dx = ...\) \[\int ax^n \, dx = \frac{ax^{n+1}}{n+1}+c\]
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]
If we differentiate \(y\) twice with respect to \(x\), what do we get? \[\frac{d^2 y}{dx^2}\]
What effect will the transformation \(y=f(-x)\) have on the graph of \(y=f(x)\)? Reflection in the \(y\) axis
What effect will the transformation \(y=-f(x)\) have on the graph of \(y=f(x)\)? Reflection in the \(x\) axis
\[\left ( \sqrt[n]{x} \right )^m=... ?\] \[\left ( \sqrt[n]{x} \right )^m=x^\frac{m}{n}\]
\[a^{-n}=...?\] \[a^{-n}=\frac{1}{a^n}\]
\[a^0=...?\] \[a^0=1\]
\[x^{\frac{1}{n}}=...?\] \[x^{\frac{1}{n}}=\sqrt[n]{x}\]
\[\left ( ab \right )^n=...?\] \[\left ( ab \right )^n=a^n b^n\]
What does the graph of \(y=\frac{1}{x}\) look like? 70d4cc6d-5ad1-490d-a0ef-78e4ae4ece26.jpg (image/jpg)
What does the graph of \(y=a^x\), where \(a>0\) look like? The \(x\) axis is an asymptote 068f39af-78c6-4e8f-aa2b-a8319aaf9bd3.gif (image/gif)
What do the graphs \(y=x^3\) and \(y=-x^3\) look like? 2ca4c950-b848-4f85-a75b-bf2e0f361e6d.jpg (image/jpg)
What does \(\sum_{r=1}^{4}a_r\) mean? \[\sum_{r=1}^{4}a_r=a_1+a_2+a_3+a_4\]
Formula for the \(n\)th term of an arithmetic sequence... [given in the formulae booklet] \[u_n=a+(n-1)d\]
Formula for the sum of the first \(n\) terms of an arithmetic sequence... [given in the formulae booklet] \[S_n=\frac{n}{2}\left ( 2a+(n-1)d \right )\] or \[S_n=\frac{n}{2}\left ( a+l \right )\] where \(l\) is the last term
If we are given \(\frac{dy}{dx}\) or \(f'(x)\) and told to find \(y\) or \(f(x)\), we need to... Integrate [remember to include \(+c\)]
Integration is the reverse of ... ? Differentiation
Differentiation is the reverse of ... ? Integration
The rate of change of \(y\) with respect to \(x\) is also called...? \[\frac{dy}{dx}\]
The formula for finding the roots of \[ax^2+bx+c=0\] \[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\]
\[a^m \div a^n = ... ?\] \[a^m \div a^n = a^{m-n}\]
\[\left (a^m \right )^n=...?\] \[\left (a^m \right )^n=a^{mn}\]
To simplify \( \frac{a}{b-\sqrt{c}} \)... (a.k.a. 'rationalising the denominator') Multiply by \[ \frac{b+\sqrt{c}}{b+\sqrt{c}} \]
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