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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.

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Plotting graphsWrite a table for the values you are going to plot – even if the question does not give you one.

x

y

𝑦=35+3cos (6 𝑥−30) 0 5 10 15 20 25

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)

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−200 200 400

−1

1

2

x

y

−2 2 4

−2

2

4

x

y

Plotting graphsTips: 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

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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)÷9Put 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

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Drawing a Gradient

0 2 41

2

3

4

x

y

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.

0 2 41

2

3

4

x

y

Like this

Not like this

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Gradients

0 2 41

2

3

4

x

y

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

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Gradients

Rise

Run

Gradient = rise run

Always remember that uphill is positive, downhill is negative

The units of the gradient are

Units of vertical axisUnits of horizontal axis

Or units of vertical axis “per” units of horizontal axisEg. 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

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GradientsThe 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 = 3 - 0.025x ?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

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−2 2 4

−2

2

4

x

y

−2 2 4

−2

2

4

x

y

Transformations: TranslationAdding a number outside the function moves the graph upwards (translates upwards in the vertical direction by that number of units)Subtracting moves it down

𝑦= 𝑓 (𝑥 ) 𝑦= 𝑓 (𝑥−2)

Subtracting a number “inside” the function moves the graph right (translates in the horizontal direction towards the right).Adding moves it left.

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−2 2 4

−2

2

4

x

y

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

−2 2 4

−2

2

4

x

y

Multiplying by a number “inside” a function squishes it horizontally (stretches it horizontally with scale factor)

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Quadratics

−2 2 4

−2

2

4

x

yC 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

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QuadraticsSolving 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…

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

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

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Sin and Cos (Period and amplitude)

The period of and is 360o

−200 200 400

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1

2

x

y

Amplitude is the vertical distance from the middle of a wave to the maximum or minimum.

The amplitude of and is 1

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.

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Sin and Cos (Amplitude after transformation)

The amplitude of is 1

If this is transformed by a vertical stretch, the amplitude is multiplied by the scale factor

(eg , the vertical scale factor is 3 and the amplitude is 3)

−200 200 400 600

−2

2

4

x

y 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

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Sin and Cos (Period after transformation)

If this is transformed by a horizontal stretch, the period is multiplied by the scale factor

(eg , the vertical scale factor is ½ so the period is 360x½=180 )

60 120 180 240 300 360

−1

1

2

x

y

Wavelength or period is the horizontal distance between equivalent points on the wave.

The period of is 360o

Period is not affected by translation in any direction or by vertical stretches

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Inverses

1

2

3

x

y

and are inverses

1

2

3

x

y

and are 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.

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Exponentials and logs

1

2

3

x

yThe graph crosses the y axis at 1, goes through the coordinate (1, e) and gets very large very fastThe graph crosses th

The graph crosses th

When x is negative, the graph gets closer and closer to 0, but never reaches it.

The graph crosses the x axis at 1, and when x is less than 1, gets very large and negative. It never reaches the y axis

Asymptote: A line that is

approached, but never reached, like for the

graph.

Exponentials and logs

The graph crosses thThe graph crosses th

Log rules: Index rules:

Examples

Plot ln(t) against xGradient = cIntercept =ln(B)

Plot ln(p) against ln(t)Gradient = kIntercept =ln(A)

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