5059110
Beschreibung
Karteikarten von Daniel Cox, aktualisiert more than 1 year ago
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Erstellt von Daniel Cox
vor mehr als 8 Jahre
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| Frage | Antworten |
| 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 \(x^2-8x+7>0\) | \(x<1\) 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, \(4!=4\times 3\times 2\times 1=24\) |
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