| 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? | 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? | 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 |
| 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? | 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 |
| 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 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 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? | 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 |
| 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 |
