# Edexcel Core 1 Maths - Key Facts

Flashcards by Daniel Cox, updated more than 1 year ago
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### Description

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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