Pure Maths

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Transformation y = |f(x)|

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Negative values of y are reflected in x-axis

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Transformation y = f(x) + a

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+a in y-axis

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52 Terms

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Transformation y = |f(x)|

Negative values of y are reflected in x-axis

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Transformation y = f(x) + a

+a in y-axis

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Transformation y = f(x + a)

-a in x-axis

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Transformation y = af(x)

All y-coordinates ×a

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Transformation y = f(ax)

All x-coordinates ÷a

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Transformation y = -f(x)

Reflection in x-axis

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Transformation y = f(-x)

Reflection in y-axis

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When is a binomial expansion valid (yr13)?

For (a + bx)n, valid when | bx/a | < 1

or |x| < a/b

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x = an as a log?

n = logax

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loga (xk) rearranged

klogax

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log(xy)

log(x) + log(y)

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log(x/y)

log(x) - log(y)

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Order of transformations

x: reflect, stretch, translate | y: translate, stretch, reflect , modulus

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π Radians in degrees

180

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Condition for increasing function?

dy/dx ≥ 0

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Condition for decreasing function?

dy/dx ≤ 0

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Condition for local minimum?

d2y/dx2 > 0

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Condition for local maximum?

d2y/dx2 < 0

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Condition for concave function?

d²y/dx² ≤ 0

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Condition for convex function?

d²y/dx² ≥ 0

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Percentage Change

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Area of circle segment

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How to integrate a parametric function?

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Differentiate

y = ln(5x)

dy/dx = 1/x

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Graph of sec x

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Graph of cosec x

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Graph of cot x

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Domain and range of sec x

Domain: x ∈ R, x ≠ any odd multiple of 90

Range: between -1 and 1

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Domain and range of cosec x

Domain: x ∈ R, x ≠ any multiple of 180

Range: between -1 and 1

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Domain and range of cot x

Domain: x ∈ R, x ≠ any multiple of 180

Range: y ∈ R

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Differentiate y = akx

dy/dx = akx k ln(a)

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nth term of a geometric sequence

arn-1

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Equation for length of a line

[ (x2 - x1)² + ( y2 - y1)² ]1/2

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Arc Length equation

arc length = 2πr × (θ/360)

if in radians then (θ/2π)

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What is ‘n’ for trapezium rule?

Number of rectangles

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If for trapezium rule 5 ordinates are used to approximate area, what would n be?

n = 4

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Area of a triangle (not 1/2bh)

½(a)(b)(sinC)

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cos rule for triangles

a² = b² + c² - 2(b)(c)(cosA)

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Integral of f’(x) × f(x)^n

f(x)^(n+1)

÷ n+1

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What makes a function a function?

If it is one-to-one or many-to-one

ie NOT one/many - many

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When sketching two ranges e.g f(x) = 5 - 2x for x < 1 and = x² + 3 for x>=1, how do you represent these inequalities?

  • Draw both graphs up to limits

  • Use a coloured in circle for =

  • Use a not coloured in circle for < or >

  • Join the circles with a straight line

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Limitations of change in sign method

  • There can be a change in sign but not root if the graph is not continuous e.g 1/x

  • There can be no change in sign but a repeated root if it touches the axis

  • There can be no change in sign because the points are too spread apart so there are e.g two roots between them

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Graph of y = ln x

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When does Newton-Raphson fail?

  • If f’(x) gives zero it means x0 (first number being inputted) is a stationary point so the gradient is zero, therefore the tangent at x0 is a horizontal line and won't cross the x-axis so no x1 is produced

  • Iterations are diverging

  • Iterations converge to another root

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When does a function have an inverse?

When the function is one-to-one

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Volume of a cone

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Volume of a cylinder

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Volume of a sphere

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Surface area of a sphere

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Types of triangle

  • Equilateral - all sides equal

  • Isosceles - two equal sides

  • Scalene - all sides are different

  • Acute triangle - all angles > 90°

  • Obtuse triangle - one angle < 90°