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5116022

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

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 \[ \frac{b+\sqrt{c}}{b+\sqrt{c}} \] |