 # WJEC Core 1 Maths - Key Facts

Flashcards by Daniel Cox, updated more than 1 year ago More Less Created by Daniel Cox over 3 years ago 93 3 0

### Description

Key points and important formulae for the WJEC 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 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 $$x^2-8x+7>0$$ 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! 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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