AS Use of Maths USE1 Revision Cards Name …………………………………………………………. This is not intended to cover all topics in USE1, but will help with some of the more common topics Add your own notes on the backs of cards, or write extra cards. 1

Plotting graphs Write a table for the values you are going to plot – even if the question does not give you one. x y Give all answers to at least 3dp unless told otherwise. You are expected to plot points to the nearest ½ square of the graph paper, so you need at least that accuracy on your calculations. Consider using the TABLE menu on your calculator to make the calculation quicker… However, do check at least one or two values “by hand” in case you make a mistake entering the equation. Tips: In AS Use questions, make sure your calculator is working in degrees when you have cos or sin functions (SHIFT, SETUP, then move down to check the Angle setting) 2

Plotting graphs Tips: Make sure you have an eraser and a pencil in the exam. Lines through the plotted points should be a single smooth line, going through all your plotted points, without “kinks” and not too thick. Your plotted points should be accurate to the nearest half square. If you think the graph might have a minimum or maximum between two of the points you have calculated, calculate a value in between, to make sure you draw the line high/low enough. Remember that quadratic graphs are symmetrical around their min/max. Remember that sin and cos graphs should be horizontal at their min/max points. Check here 3

Plotting graphs using graphical calc Put brackets around negative numbers, For example, (-5) 2 Put brackets around any calculation on the top or bottom of a fraction, for example, (X+3) ÷9 Put brackets around the calculation “inside” a sin or cos Remember your calculator can only calculate values for things it can “see”. Use V-Window to make sure you are looking at the right part of your graph. When typing on your calculator… When you are looking for features on a graph on your calculator If you have any spare time, check a few values by hand to make sure 4

Drawing a Gradient Draw a gradient to touch the graph at the right place. Adjust the slope of the gradient so that it matches the slope of the graph for the same distance each side of the point. Like this Not like this 5

Gradients To calculate the gradient of the tangent, create a right angled triangle using the gradient line. Tips: Choose the position of your triangle so that you can accurately calculate the coordinates of its corners (ideally where the gradient crosses a major gridline in one or both directions). The triangle does not have to include the point where the tangent meets the graph. Ideally the triangle will be as large as possible – to improve accuracy. Rise Run Using the units of the horizontal and vertical axes, calculate the lengths of the rise and run. Gradient = rise run Always remember that uphill is positive, downhill is negative 6

Gradients Rise Run Gradient = rise run Always remember that uphill is positive, downhill is negative The units of the gradient are Units of vertical axis Units of horizontal axis Or units of vertical axis “per” units of horizontal axis Eg. Miles per hour The gradient represents the rate of change of the vertical units. Eg. When the horizontal value increases by 1 unit, the vertical value increases/decreases by the number of units represented by the gradient. Example: As the baby increases in length by 1cm, its weight increases by (gradient) kg 7

Gradients The equation of a straight line is y=mx+c If you have an equation and want to plot the graph, create a table and calculate at least 3 points (ideally spread across the range of the horizontal axis) NOTE: Gradient is just the number, not the x What is the gradient of y = x ? What are the coordinates of where the line meets the y axis? If you have a plotted line and you want to find the equation, create a triangle using two points on the line (where you know the coordinates accurately). Calculate rise over run to find the gradient = m. Then read off the intercept of the line with the vertical axis = c. Use the symbols of the vertical and horizontal axes instead of y and x, and the values of m and c you have calculated. The gradient is the number multiplying the x (or whatever letter is on the horizontal axis) The intercept is where the line meets the vertical axis 8

Transformations: Translation Adding a number outside the function moves the graph upwards (translates upwards in the vertical direction by that number of units) Subtracting moves it down Subtracting a number “inside” the function moves the graph right (translates in the horizontal direction towards the right). Adding moves it left. 9

Transformations: Stretch Multiplying by a number outside the function stretches it vertically with the number as the scale factor. 10

Quadratics C is the intercept with the y axis If you can factorise, roots are where one of the factors = 0 Min or max point at (-q, r) Completed square form 11

Quadratics Solving an equation means finding the value or values of x (or whatever letter) when the equation is true. When a graph crosses the x axis, y=0. Find this point using ROOT on Gsolve on your calculator. Or, use the Polynomial solver on your calculator, Or work algebraically and use the quadratic formula… When a graph crosses the y axis, x = 0. You can normally substitute this value in… Sketching a graph means showing its general shape, labelling where it crosses the axes (if it does), and labelling the coordinates of any min or max – and other points of interest Solving simultaneous equations: find the values of x and y that make both equations true at the same time. (plot both graphs and find where they intersect(cross) You can use ISCT in Gsolve 12

Sin and Cos (Period and amplitude) Amplitude is the vertical distance from the middle of a wave to the maximum or minimum. Wavelength or period is the horizontal distance between equivalent points on the wave. The steepest part of a sin or cos curve is half way between the max and min. 13

Sin and Cos (Amplitude after transformation) Amplitude is the vertical distance from the middle of a wave to the maximum or minimum. Amplitude is not affected by translation in any direction or by horizontal stretches 14

Sin and Cos (Period after transformation) Wavelength or period is the horizontal distance between equivalent points on the wave. Period is not affected by translation in any direction or by vertical stretches 15

Inverses Inverse functions “undo” each other. The graphs of inverse functions are reflections in the line y=x If you need an algebraic inverse, swap x and y and rearrange. 16

Exponentials and logs 17

Exponentials and logs Examples Plot ln(t) against x Gradient = c Intercept =ln(B) Plot ln(p) against ln(t) Gradient = k Intercept =ln(A) 18