To find the equation of a tangent, first, find the gradient of the radius, for example, if the radius went from the origin to (3,4), the gradient would be 4-0/3-0 =4/3 because the gradient = change in y/ change in x. Then find the gradient of the tangent by taking the negative reciprocal of the first gradient which is – 3/4, Now complete the rest of the equation, as the straight-line equation is y = -3/4x + c, and as y = 4 and x = 3, this = 4 = -9/4 + c, and as 25 the diameter, y = -3/4x + 25/4

What are rational numbers

A rational number is any number that has a limited number of decimal places that can be written as a fraction such as integers, fractions, terminating decimals terminating (limited amount of numbers after the decimal) and recurring decimals

How do you calculate the instantaneous rate of change?

To calculate the instantaneous rate of change by drawing a tangent to the existing curve, and making the tangent form a right-angled triangle to calculate the gradient of the point where the tangent touches the curve, (in the centre of the tangent/curve.

What are irrational numbers

Irrational numbers are numbers that continue forever without recurring numbers, examples include pi and surds e.g. square root 2

What are geometric sequences involving surds (square roots)?

Geometric sequences involving surds are sequences where the common ratio is a square root and the next terms if found by multiplying the preceding term by the square root

What does a sketched quadratic graph show?

A sketched quadratic graph shows the key points of a quadratic function such as the roots or x-coordinates where the function intercepts the x-axis, the y-intercept, and the vertex or the turning point which is the minimum or the maximum value at the top and bottom of the u’s/n’s.

What are the two steps to expanding triple brackets?

To expand three brackets expand the first two brackets, then expand that answer with the third bracket

What is the quadratic formula?

The quadratic formula is x = -b + or – square root b2– 4ac / 2a (where -b =/- is on the fraction line)

What is the quadratic equation for standard form?

The quadratic equation for standard form is ax2 + bx + c = 0

How do you use the quadratic formula to solve quadratic equations?

To use the quadratic formula, write in the a, b, and c values in the quadratic formula, from the given equation such as 2x2 + 3x – 2 = 0, in which a = 2, b = 3, and c = -2, so x = -b + or – square root b2– 4ac /2a becomes x = -(3) + or –square root 32– 4(2)(-2) /2(2) which means = -3 +/- |square root 9 +16|/ 4 and as 9+16=25 and the square root of 25 is 5 the equation = -3+/- 5/ 4. Which is then broken into two equations because of the +/- symbol: x = -3 + 5 / 4 and – 3 – 5 / 4 and as – 3 + 5 = 2, and – 3 – 5 = =8 the equations become 2/4 and -8/4 which simplify to ½ and -2 so x = ½ or x = -2

What is the inter quartile range?

The inter quartile range is the range from the first quartile range (value) to the last/third quartile range (value) IQR = range from Q1 – Q3

How do you turn a fraction into a recurring decimal?

To turn a fraction into a recurring decimal multiply the denominator so it has 9's in it, then multiply the nuetortor by the same number. e.g. 32/666 x 1.5 = 48/999, and because the numerator needs to have the same number of digits as the denominator, a 0 is before 48 so it becomes 048/999 which gives the recurring decimal 0.048 (where a dot is over the 0 and 8)

What is the surface area of a cuboid?

Becuase cuboids have 3 types of faces and 2 of each Surface area of a cuboid = 2lw x 2lh x 2hw

What is the area of a sector?

The area of a sector is r^2 x a/2 where a is the interior angle

How do you do negative and fractional indices?

To do negative and fractional indices, when the number is whole put 1 over it and if the number is a fraction, flip it And in both cases the negative indices then become positive and the expressions are simplified And if an indice is negative and fractional first flip the fraction number then square root it

How do you simplify surds?

To simplify surds find the two highest common factors also in surd form – then undo the square root that has an integer answer, put it in front of the other surd, and simplify

How do you find the equation of a tangent?

What is the equation for interior angles?

Interior angles = 180 (n-2), when n = number of sides

What is the equation for exterior angles?

The equation for exterior angles is 360/n, where n = number of sides, and all exterior angles = 360

What is the formula for arc length?

The formula for arclength is L/ 0 = C / 2pi, where L = length and 0 = angle

What is 1kg in Pounds?

1kg in Pounds is 2.205 pounds

What is 1L in m^3

1L in m^3 is 0.001m^3

How are functions calculated with specific values?

A function would be calculated when given a specific value for example f(4) and an equation such as f(x)=3x + 2, by substituting 4 in the place of any x’s in the function, so for this equation f(x) = 3x + 2 is f(4) = 3x4 + 2 = 14

How are functions calculated with specific values?

What are the three things you need to find the equation of a tangent?

What are the three things you need to find the equation of a tangent? To find the equation of a tangent you need the value of x, the value of y and the gradient,

How do you find the equation of a tangent?

How do you calculate the instantaneous rate of change?

What does a sketched quadratic graph show?

What needs to be done to sketch a quadratic graph?

To sketch a quadratic graph identify the coefficient of x^2, which tells you if the graph is u or n shaped, and identify the constant term, which tells you the y-intercept. Now set y = 0 and solve the resulting quadratic equation to find the roots, Then complete the square to find the coordinates of the vertex and sketch the graph with labelled key points.

When will the quadratic be u shaped or n shaped?

The quadratic will be u shaped when the coefficient of x^2 is positive, and the quadratic will be n shaped when the coefficient of x^2 is negative.

What is the general equation for quadratic graphs?

The equation for quadratic graphs is ax^2 + bx + c

What do you do to plot a quadratic graph?

To plot a quadratic graph draw a table for the values of x between -2 and 3 and the corresponding y values. Then plot these coordinates and join them with a smooth u or n curve

Why would the quadratic need to be expressed by completing the square?

The quadratic would need to be expressed by completing the square to help solve the quadratic equation and to find the coordinates of the minimum or maximum point of a quadratic graph

How is the quadratic formula expressed for completing the square?

For completing the square the quadratic formula is expressed from ax^2 + bx + c to a (x + d)^2 + e, where d = b/2a, and e = c – b^2/4a, although it may also be expressed as e = c – ad^2

How do you estimate the area under a curve using triangles and trapeziums?

You estimate the area under a curve using triangles and trapeziums, by splitting the graph vertically so that each value X becomes a trapezium or triangle, and then calculating the area of the shapes with their given coordinates, e.g. as the area of a triangle is ½ x base x height, if the triangle’s base is 1, and its height is 4.4, the equation becomes ½ x 1 x 4.4 = 2.2, which is added to the other shapes areas

Why are graphs with a curved line along both axis called cubic graphs?

Graphs with a curved line along both axis are called cubic graphs because their equations always contain an x^3 term.

What is the general equation for cubic graphs?

The general equation for cubic graphs is ax^3 + bx^2 + cx + d, where a is another number above 0, and b, c, and d are any number including 0

How can you tell the direction of a cubic graph from its equation?

You can tell the direction of a cubic graph from its equation because positive x^3 values represent a line that goes from the bottom left to the top right (or up), and negative x^3 values represent a line that goes from the top left to the bottom right (or down)

What needs to be done to plot a cubic graph?

To plot a cubic graph fill in an x value for y value table, in accordance with the equation given cubic equation e.g. y = x^3 + x – 2 Then plot all those points on the graph and join them up together in a curved line

What are reciprocal graphs?

Reciprocal graphs are graphs where y = A / x, where A is any value

How are reciprocal graphs drawn?

Reciprocal graphs are drawn with one branch in the top right column and another in the bottom left column, so they are symmetrical, which can be checked by drawing the line y = x

How can you check if a graph is y = 1/x?

You can check if a graph is y = 1/x by making a table of x values to find the corresponding values in the equation y = 1/x

Why can x = 0 not be plotted on a reciprocal graph?

X = 0 cannot be plotted on a reciprocal graph because if it were in the equation y = 1/x 1/0 is infinite, causing there to be no way to plot it on a graph, because a curve doesn’t cross the y -axis, yet it vertically down the middle

What do reciprocal graphs look like when the value of A is bigger than 1? When the value of A is between 0 and 1? And when the value of A is negative?

When the value of A is bigger than 1, the branches move outwards, away from the axis (centre) When the value of A is between 0 and 1, the branches move inwards towards the axis (centre) When the value of A is a negative number, the branches swap columns so they are in the top left and bottom right instead

What is the general equation for exponential graphs?

The general equation for exponential graphs is y = k^x

How are exponential graphs drawn?

Exponential graphs are drawn with a branch that begins vertical/on the axis, and then curves upwards

What is the value of k?

The value of k is the value of y at x = 1

In algebraic proof is 2n odd or even?

In algebraic proof, because n is a number, 2n must always be even

Is 2n + 1 odd or even?

Because 2n is always even, 2n + 1 Is always odd

When a triangle is labelled a, b, c, how can you prove it has a sum of 180 degrees?

You can prove that a triangle has a sum of 180 degrees when labelled a, b, c, by drawing one of the bottom exterior angles with a line going through it and labelling these two angles x and y, on a straight line, now state that the angle at the top of the triangle is alternate to x, making the other bottom angle corresponding to y, and as c + x + y = 180 degrees, being on a straight line, a + b + c = 180 degrees

What are the two methods of algebraic proof?

The two methods of algebraic proof are counter examples and identity symbols

How could a counter example be used in algebraic proof?

A counterexample could be used in algebraic proof if the question asked to disprove: all prime numbers are odd, by saying this statement is false because 2 is a prime number and 2 is even not odd.

How can you prove the statement: ‘the difference between two prime numbers is always even’, with a counter example?

If the question asked: prove the statement is wrong: ‘the difference between two prime numbers is always even’, a counter example that could be used is the difference between 2 and 5 is 3, an odd number, so the difference between two prime numbers is not always even.

How are identity symbols used to prove algebraic identities are true?

Identity symbols are used to prove algebraic identities are true, by making the left side of the equation look the same as the right because terms cannot be moved from one side of the equation to the other

What are the steps for making the left side of the equation look the same as the right?

To make the left side of the equation look like the right, on the left side, expand the brackets, then collect like terms, and if the right side is factorized, factorize the left side to look the same.

If equations have an equals sign instead of an identity sign what changes?

If equations have an equals sign instead of an identity sign nothing changes, still make the left side look the same as the right

What does y = f(x) + a mean? what does y = -f(x) mean? what does y = af(x) mean?

Y = f(x) + a means the curve shifts up or down y = -f(x) means the curve is reflected in the x-axis y = af(x) means the curve is vertically squashed or stretched

When will a curve be stretched?

A curve will be stretched when the a in the equation y = af(x), is any number bigger than 1, e.g. if the equation was y = 2f(x) every y value would need to be multiplied by 2/ on the left and right axis.

When will a curve be squashed?

A curve will be squashed when a in the equation y = af(x) is any number smaller than 1 such as a fraction, e.g. is the equation was y = 1/3f(x) the curve would be squashed by a 1/3

How do you calculate simultaneous equations?

To calculate simultaneous equations eliminate one variable in both equations by multiplying by a common factor then subtracting the first equation from the second and substituting the answer in one of the two equations to find the other variable

How do you substitute non-linear simultaneous equations?

To substitute non-linear simultaneous equations rearrange the linear equation to get one unknown value behind the equals sign, and the other in front, Then substitute the linear equation into the non-linear equation Expand and solve the new quadratic formed And substitute both values back into the simplest equation to find the variables in the other equation

What do you do to graphically solve simultaneous equations?

To graphically solve simultaneous equations plot the equations on the graph and see where they intersect, to find the x and y value

What are the three methods for solving simultaneous equations?

The three methods for solving simultaneous equations are plotting them on a graph, substitution, and elimination, which are both when one equation is rearranged and combined with the other equation, to find the x and y values.

What does the equation y = 2x – 1 mean?

The equation y = 2x – 1 means x = 2, the equation would become y = 2 x 2 – 1, and solved so y = 3, when x = 2

How would the straight-line equation y = 2x =- 1 be plot on a graph?

The straight-line equation y = 2x – 1 would be plot on a graph by drawing a line, where every value of y is one more than the value of x

What does the equation y + 2x = 5 mean?

The equation y + 2x = 5 means if x = 0, and y = 5, the equation would become y + 2 x 0 = 5, and because 2 x 0 = 2, y = 1, and x = 2

What does the equation f(x) = y stand for?

The equation f(x) = y stand for function(input) = output, where f is not multiplied by x but is simply a label

How are f(x) and y interchangeable?

In functions, the input is always before the equals sign and the output can be represented by both f(x) and y, E.g. f(x) = y is the same as y = f(x)

How are functions calculated with specific values?

What are the terms of the input and output in functions?

The term for the input is the domain and the term for the output is the range

How are functions plotted on a graph?

Functions come in equations that can be plotted as coordinates by regarding the input and output values as ordered pairs (x,y)

What does the vertical line test, test for?

The vertical line test that tests if a graph has more than one output for a single input which would disqualify is as a function

How is the vertical line test done?

The vertical line test is done plotting every y and x value for a function and then drawing a vertical line down a graph. The line is moved left and right to see the point where the vertical line intersects with the graph. If the vertical line only intersects at exclusively one point for every value of x in the graph, there is only one output value for every input value, so the graph qualifies as a function.

Why does y^2 = x not qualify as a function?

Y^2 = x does not qualify as a function because it does not pass the vertical line test, in the equation y^2 = x, the right x-axis intersects with the vertical line in two places, so this equation has multiple outputs for its single input.

What is a composite function?

A composite function is when one function is put into another function.

How can one function be put into another function?

If the question asks what f g(x) is, you put one function into another function, to do this find the input (x) Such as x in the equation f (x) = 2x + 1, and then put the second equation of g (x) = x^2 + 1 into the input, so the equation becomes f g (x) = 2 (x^2 + 1) + 1. Now solve and simplify the resultant equation

What do you do if a question asks to find g f(x) in simultaneous equations?

If a question asks to find g f(x) in simultaneous equations put the input of the first equation into the second equation, then solve and simplify it

What are inverse functions written as?

Inverse functions are written as f ^-1 (x)

What is the purpose of inverse functions?

The purpose of inverse functions is to give the input given any output, whereas normal functions gives the output given any input

What is f (x) = 2x - 4?

F (x) = 2x - 4 is rearrange to find the input, by letting f (x) = y, This is done by rearranging to y = 2x - 4, then add four: y + 4 = 2x and divide by 2 to make x the subject y + 4 / 2 = x now because the output of y becomes the input y + 4 / 2 = x becomes f^-1 (x) = x + 4 / 2

What is the question: prove ff^-1 (x) = x asking for?

The question: prove ff^-1 (x) = x asks to find the inverse function and then input that answer into the original function

What are arithmetic sequences?

Arithmetic sequences are when the common difference in the sequences is an addition or subtraction

What are geometric sequences?

Geometric sequences are when the common ratio is a division or multiplication, e.g. 3, 6, 12, 24, that has a common ratio of 2 because 6/3 and 24/12 = 2

How do you find the common ratio in divisions?

To find the common ratio in divisions find the number that is being multiplied each time, e.g. if the common ratio was dividing by 3 each time, it is being multiplied by 1/3

How can you check the common ratio?

You can check the common ratio by dividing the next term by the previous term.

What do you do to work out the quadratic formula without a calculator?

To work out the quadratic formula without a calculator factorise the quadratic

How do you factorise quadratics?

Factorise quadratics by writing out possible factor pairs that multiply to make the integer term, then find which pair can be added to make the x term, and put these numbers into the brackets

How do you convert m^2 to cm^2?

To convert ^2 to cm^2, when given the area, find the numbers that multiply to make the area such as 3m^2 = 1m x 3m convert them to cm = 100cm x 300cm and their sum: 30,000cm^2 is the conversion.

What are the laws of indices?

1^2 x 1^5 – same base – add 1^2 / 1^5 – same base – subtract (2x^5) 4 – integer inside also multiply by Indice 3^2 x 2^-5 = 6^-3 – multiply then add

How do you do negative and fractional indices?

How do you express 8^k x 3^3k, as a power of 6?

Because 8^k and 3^3k do not share the same integer they must be changed to share the integer make them both either X^k or X^3k And because 8 = 2^3, 8^k becomes 2^3k x 3^3k Now because they have the same integer they can be multiplied and = 6^3k – a power of 6

If the compound interest per annum is 5% what do you multiply the input by?

If the compound interest per annum is 5% multiply the input by 1.05 because 100% = 5% = 105%

How do you enlarge by scale factor 2, without given coordinates?

To enlarge by scale factor 2, work out starting coordinates x 2 E.g. if the corner of a right-angled triangle is 3,1 enlarging by scale factor 2 means it would start from 6,2 and be 2x the size of shape A

What do you to do enlarge by scale factor ½?

To enlarge by scale factor ½ halve all the coordinates

How do you find the centre of enlargement?

To find the centre of enlargement Find the number of squares along the x-axis then up the y-axis the given centre is away from the shape A’s centre. Now multiply/divide this if the scale factor is an integer/fraction And the new number of squares is how many the coordinates will move along the -axis and up the y-axis to reach the starting centre of shape B

What type of triangle are sine and cosine rules used on?

Sin and Cosine rules are used on any triangle that’s not a right angle

How does the sine rule work, and when is the cosine rule used?

When using the sine rule all the angles and sides are labelled A, B and C, and a,b, and c, in which every angle is opposite to its side – of a, b, and c. And it is used when one pair of an angle and side are known e.g. A and a, but only half of another is known e.g. B, to find the other half of b. While the cosine rule is used to find an angle when all three sides are known or when two sides and their included angle is known.