Algebra and graphs flashcards

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Expand and simplify \((x+3)(x-5)\).
Multiply every term: \(x^2-5x+3x-15 = x^2-2x-15\).
How do you factorise a quadratic \(x^2+bx+c\)?
Find two numbers that multiply to \(c\) and add to \(b\). E.g. \(x^2+7x+12=(x+3)(x+4)\) because \(3\times4=12\) and \(3+4=7\).
Factorise \(x^2-49\) and name the pattern.
Difference of two squares: \(a^2-b^2=(a-b)(a+b)\), so \(x^2-49=(x-7)(x+7)\).
(Extended) How do you factorise \(2x^2+7x+3\)?
Find two numbers multiplying to \(2\times3=6\) and adding to 7 (6 and 1), split the middle term: \(2x^2+6x+x+3=2x(x+3)+1(x+3)=(2x+1)(x+3)\).
State the three index laws for multiplying, dividing and raising a power to a power.
\(a^m\times a^n=a^{m+n}\); \(a^m\div a^n=a^{m-n}\); \((a^m)^n=a^{mn}\). Also \(a^0=1\) and \(a^{-n}=\frac{1}{a^n}\).
(Extended) Evaluate \(27^{\frac{2}{3}}\).
The denominator is the root, the numerator the power: \(27^{\frac23}=(\sqrt[3]{27})^2=3^2=9\).
How do you rearrange a formula to make a different letter the subject?
Treat it like an equation: undo operations in reverse order, doing the same to both sides. Collect terms in the new subject on one side and factorise if it appears more than once.
Make \(x\) the subject of \(y=\frac{3x-2}{5}\).
Multiply by 5: \(5y=3x-2\). Add 2: \(5y+2=3x\). Divide by 3: \(x=\frac{5y+2}{3}\).
Solve the simultaneous equations \(x+y=7\), \(2x-y=2\).
Add the equations to eliminate \(y\): \(3x=9\), so \(x=3\). Substitute: \(y=4\). Check in \(2x-y=6-4=2\), correct.
(Extended) State the quadratic formula and when to use it.
For \(ax^2+bx+c=0\): \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). Use it when the quadratic will not factorise easily.
Solve \(x^2-3x-10=0\) by factorising.
\((x-5)(x+2)=0\), so \(x=5\) or \(x=-2\).
(Extended) Write \(x^2+6x+1\) in the form \((x+p)^2+q\).
Halve the coefficient of \(x\): \(p=3\). Then \(x^2+6x+1=(x+3)^2-9+1=(x+3)^2-8\).
Solve the inequality \(3-2x>9\) and state the key rule for inequalities.
\(-2x>6\), so \(x<-3\). Multiplying or dividing both sides by a negative number reverses the inequality sign.
How do you find the \(n\)th term of a linear sequence such as 5, 8, 11, 14?
The common difference \(d=3\) gives \(3n\); compare with the first term: \(5-3=2\). So the \(n\)th term is \(3n+2\).
(Extended) Find the \(n\)th term of the quadratic sequence 3, 8, 15, 24.
Second difference is 2, so the \(n^2\) coefficient is \(\frac22=1\). Subtract \(n^2\): 2, 4, 6, 8, which is \(2n\). So the \(n\)th term is \(n^2+2n\).
(Extended) If \(f(x)=2x+3\) and \(g(x)=x^2\), find \(fg(x)\).
\(fg(x)=f(g(x))\): apply \(g\) first, then \(f\). \(fg(x)=2x^2+3\).
(Extended) How do you find the inverse function \(f^{-1}(x)\) of \(f(x)=\frac{x+4}{3}\)?
Write \(y=\frac{x+4}{3}\), swap \(x\) and \(y\), and solve for \(y\): \(x=\frac{y+4}{3}\Rightarrow y=3x-4\). So \(f^{-1}(x)=3x-4\).
What do \(m\) and \(c\) represent in \(y=mx+c\)?
\(m\) is the gradient (steepness) of the line and \(c\) is the \(y\)-intercept, where the line crosses the \(y\)-axis.
What shape is the graph of \(y=ax^2+bx+c\) and what does the sign of \(a\) tell you?
A parabola. If \(a>0\) it opens upwards (U shape, minimum point); if \(a<0\) it opens downwards (\(\cap\) shape, maximum point).
Describe the shape of the graph of \(y=\frac{1}{x}\).
A hyperbola in two branches, never touching the axes. The \(x\)-axis and \(y\)-axis are asymptotes. The graph is undefined at \(x=0\).
(Extended) What is the general shape of \(y=a^x\) for \(a>1\), and what does it pass through?
Exponential growth: it rises ever more steeply, always above the \(x\)-axis (which is an asymptote), and passes through \((0,1)\).
(Extended) How do you find the gradient of a curve at a point on a graph?
Draw a tangent to the curve at that point, then find its gradient using \(\frac{\text{change in }y}{\text{change in }x}\) between two points on the tangent.
(Extended) How do you differentiate \(y=ax^n\)?
Multiply by the power and reduce the power by 1: \(\frac{dy}{dx}=anx^{n-1}\). E.g. \(y=4x^3\) gives \(\frac{dy}{dx}=12x^2\).
(Extended) Find the \(x\)-coordinates of the turning points of \(y=x^3-12x\).
Set \(\frac{dy}{dx}=3x^2-12=0\), so \(x^2=4\) and \(x=\pm2\).
(Extended) How can you tell whether a turning point is a maximum or minimum?
Find \(\frac{d^2y}{dx^2}\) at that point: positive means a minimum, negative means a maximum. Alternatively check the sign of the gradient either side.
What is the difference between direct and inverse proportion?
Direct: \(y\) increases as \(x\) increases, \(y=kx\). Inverse: \(y\) decreases as \(x\) increases, \(y=\frac{k}{x}\). Find \(k\) from known values.
(Extended) \(y\) is proportional to \(x^2\), and \(y=50\) when \(x=5\). Find \(y\) when \(x=8\).
\(y=kx^2\): \(50=25k\), so \(k=2\). Then \(y=2\times64=128\).
(Extended) Simplify \(\frac{x^2-9}{x^2+3x}\).
Factorise: \(\frac{(x-3)(x+3)}{x(x+3)}\). Cancel the common factor \((x+3)\) to get \(\frac{x-3}{x}\).
(Extended) Solve \(2x^2-5x-1=0\) giving answers to 2 decimal places.
\(x=\frac{5\pm\sqrt{25+8}}{4}=\frac{5\pm\sqrt{33}}{4}\), so \(x=2.69\) or \(x=-0.19\).

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