What is log a a equal to?

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Cora Angela Allison
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2 How would you differentiate a function like y = sin ax? What is log a a equal to? How do you prove three 3-D points are collinear? What is the general equation of a straight line passing through (a,b) with gradient m? How do you draw the graph of -f(x) given the graph of f(x)? What is the turning point of y = b - 3(x - a)²? Max or min? Given an equation like m = m o e -3k and an amount by which it has been decayed, how do you find k? How do you solve an equation of the form acosx+bsinx+c=0? How would you differentiate a function like y = sin 3 x? How do you solve equations of the form 3 x = 0.155?

3 (a,b) Maximum dy/dx = acos ax (i) Fit in m and m o (ii) Rearrange to get e 3k = (iii) Take logs (iv) Solve 1 (i) Change acosx + bsinx into Rcos(x- (ii) Rearrange and solve Prove they are the same vector multiplied by different or the same numbers (i) Write as (sin x)3 (ii) Multiply by the power (iii) Reduce power by 1 (iv) Multiply by the derivative of the bracket y - b = m(x - a) (i) Take logs (ii) Bring x down to the front (iii) Solve as normal Reflect the graph in the x-axis

4 where a curve crosses the y-axis? Name the steps you take in order to differentiate functions like x f(x) 3x 1 x What is the condition for no real roots? What is the cosine rule used to find a missing angle? What is the cosine rule used to find a missing side? What is the gradient of a horizontal line? the angle between two vectors? What do you know about the gradient of two perpendicular lines? What is the equation of a horizontal line passing through the point (a,b)? the gradient of a line, given the angle the line makes with the positive direction of the x-axis?

5 0 Let x = 0 and solve cos a.b a b (i) Change roots to powers (ii) Split up into 3 fractions (iii) Simplify each term (iv) Differentiate They multiply together to give -1 b 4ac 0 y = b cosa b c bc a m = tan a b c bccosa

6 Given experimental data, how do you find an equation in the form y = ab x or y = ax b? What is log a x + log a y equal to? How do you solve an equation of the form 3sinx + 1 = 0? What does a.a equal? where a line and a circle intersect? For what values is the function f(x) x - a undefined? What do you get when you differentiate sin x? What do you get when you differentiate cos ax? What is log a x n equal to? How do you factorise a cubic expression like x 3 - x - x +?

7 log a xy cos x (i) Take logs (ii) Rearrange to get a straight line equation (iii) Determine type (iv) Find solution (i) Rearrange to sinx=. (ii) Decide on quads (iii) Ignore any neg and press INV sin to get angle (iv) Work out two answers -asin ax a n log a x (i) Rearrange line to get x =. or y = (ii) Fit into circle and solve Synthetic division (nesting process) using factors of last number x < a

8 How do you draw the graph of f(x) given the graph of f(x)? How do you calculate the area between two curves? What is an altitude of a triangle? What is the condition for equal roots? What is the quadratic formula and explain when it is used What is a median of a triangle? If f(t) is the distance travelled in a certain time t seconds, then what does f (t) represent? If f(x) = x n then what is f (x)? How do you change from degrees to radians? the exact values of sinx or tanx given cosx = a/b?

9 b -4ac=0 Double the y-coordinates x b b a It is used to find roots of an equation when it does not factorise A line from a vertex to the middle of the opposite side 4ac Divide by 180 and multiply by π (i) Let equations equal each other (ii) Solve to find limits (ii) Integrate top-bottom functions between limits A line from a vertex perpendicular to the opposite side Speed (Velocity) (i) Draw triangle (ii) Use Pythagoras to fill in missing side (iii) Read values off the triangle using SOHCAHTOA nx n-1

10 Do you know you table of exact values? How do you add or subtract vectors? b a x 3 dx If u = ai+bj+ck then what is u in component form? How do you prove that a line is a tangent to a circle? a If u b c then what is u? How do you multiply a vector by a number? What is the centre and radius of a circle in the form (x-a) +(y-b) =r? What is the centre and radius of a circle in the form x +y +gx+fy+c=0

11 sin cos tan o o o u a b c Add or subtract matching components (i) Rearrange line to make y= or x= (ii) Fit line into circle (iii) Prove it has equal roots b-4ac=0 or find the repeated roots (i) Integrate (ii) Fit in b and a and subtract Multiply each entry by the number Work out length a b c Centre (-g,-f) Radius g f c Centre (a,b) Radius = r

12 the value of a in the polynomial x 3 +ax +4x+3 given a factor of the polynomial or the remainder when the polynomial is divided by a number? What is log a 1 equal to? (ax b) dx the midpoint of a line joining two points? f(g(x)) for given functions f(x) and g(x)? x 1 x dx What do you know about the gradient of two parallel lines? How do you differentiate a bracket without multiplying it out? What is the condition for real roots? the equation of a perpendicular bisector of a line?

13 0 (i) Do synthetic division (ii) Let the expression = 0 or the remainder and solve the equation They are the same (i) Increase power by 1 (ii) Divide by new power (iii) Divide by the derivative of bracket (i) Multiply by power (ii) Reduce power by 1 (iii) Multiply by derivative of bracket Add the coordinates and divide by two x1 x y1 y, b -4ac 0 Fit g(x) into f(x) i.e. each x in f(x) is replaced by the function g(x) (i) Find the midpoint of the line (ii) Find the gradient of the line (iii) Find the gradient perpendicular to the given line (iv) Use midpoint and gradient in y-b=m(x-a) (i) Change root to power (ii) Split up into fractions (iii) Simplify each term (iv) Integrate each term (v) REMEMBER +C!

14 Which points does the graph y=log a x always pass through? the inverse of a function? Given a rec. relation in the form u n+1 =au n +b and 3 consecutive terms, how do you find the values of a and b? How do you know when a recurrence relation of the form u n+1 =au n +b converges? the rate of change of a function at a particular point? the equation of a tangent to a curve at the point when x=a? What is the turning point of y=(x-a) + b? Max or min? How do you complete the square for functions like x + 1x + 3? How do you draw a graph of the form y = cosx+a or y = sinx+a? If y is the equation of a curve, what is represented by dy/dx?

15 (i) Differentiate (ii) Fit a into f (x) to get gradient (iii) Fit a into f(x) to get coordinate (iv) Use y-b = m(x-a) (1.0) and (a,1) (a,b) Minimum (i) (x +6x) +3 (ii) (x +6x+(6/) ) +3-(6/) ) (iii) (x +6x+9) (iv) (x+3) - 15 Move the graph up a units Gradient (i) Change f(x) into y (ii) Rearrange equation to x =.. (iii) Change x to y and y to x (i) Fit 1st term into u n and nd term into u n+1 (ii) Fit nd term into u n and 3rd term into u n+1 (iii) Solve simultaneous equations -1 < a < 1 (i) Differentiate (ii) Fit in given x value

16 How do you integrate cos ax? What is a unit vector? What is the centre and radius of a circle in the form x + y = r How do you calculate the area under a curve? the equation of a tangent to a circle at a particular point on the circumference? exact values of sinx and cosx given cos x = 3/5? How do you solve equations like cosx-5sinx-1=0? (0 x How do you solve quadratic inequations like x + 5x - 6 0? How do you name the angle between a line and a plane? a unit vector parallel to a given vector?

17 (i) Find centre (ii) Find gradient from centre to point (iii) Find perpendicular gradient (iv) Use y-b=m(x-a) 1 a sin ax (i) Draw a triangle (ii) Find missing side (iii) Expand double angle (iv) Fit in values from Δ A vector of length 1 unit (i) Fit in 1-sin x (ii) Factorise (iii) Solve equations Centre (0,0) Radius = r (i) Factorise (ii) Draw graph (iii) Read values above the x-axis (i) Find the length of the given vector (ii) Divide all the components by this length (i) Integrate (ii) Fit in two limits and subtract to find area (i) Start at end of the line (ii) Go to where the line meets the plane (iii) Go to the point on the plane directly under the start of the line

18 where a curve is decreasing? How do you integrate sin ax? How would you find the maximum or minimum value of a function given its equation? How do you show that a curve is always increasing? How do you show that a curve is always decreasing? How do you integrate x n? Who loves maths? the limit of a recurrence relation from its formula? What is an equivalent expression for y = a x using logs? the maximum or minimum values of acosx + bsin x + c?

19 1 a cos ax (i) Differentiate (ii) Let f (x)=0 (iii) Solve to find st. pts. (iv) Draw nature table (v) Read values for graph decreasing (i) Differentiate (ii) Show that f (x) is a square (i) Differentiate (ii) Let f (x)=0 (iii) Solve to find st. pts. (iv) Draw nature table (v) Read off max or min x n 1 n 1 c (i) Differentiate (ii) Show that f (x) is neg and square e.g. -(x+a) x = log a y ME!!!!! (i) Change acosx+bsinx into Rcos(x-α) (ii) Max is R + c (i) Let u n+1 = u n = L (ii) Solve equation

20 How do you solve equations like sin x = 0.5? (0 x How do you draw a graph of the form y=cos(x+a) or y=sin(x+a)? How do you solve equations of the form sin x = 0.5? (0 x How do you draw a graph of the form y=cosbx or y=sinbx? the root of an equation between two given values to 1 dp? For what values of a function is the function said to be undefined? How do you show that a root of an equation lies between two given values? How do you draw a graph of the form y=acosx or y=asinx? How do you solve equations of the form sinx = 0.5? (0 x where a curve crosses the x-axis?

21 When you fit in a value of x and cannot get an answer Fit in two values and show one is positive and one is negative Draw normal y=cosx or y=sinx graph with maximum a and minimum -a (i) Decide on the quadrants (sin is +) (ii) Ignore the sign and press INV sin to get angle (iii) Work out your answers Move the graph of y=cosx or y=sinx a units to the LEFT (i) Rearrange to get sin x =.. (ii) Find answers in all 4 quadrants i) Decide on the quadrants (sin is +) (ii) Ignore the sign and press INV sin to get angle (iii) Work out your angles (iv) Divide by for answers Draw the normal graphs but fit in b waves between 0 and 360º Let y = 0 or f(x) = 0 and solve for x Iteration

22 Which points does the graph y = a x always pass through? the median AM of triangle ABC? What is sin x + cos x equal to? What do you get when you differentiate cos x? How do you show that two vectors are perpendicular? How do you change from radians to degrees? How do you draw the graph of f (x) given the graph of f(x)? What is the equation of a vertical line passing through point (a,b)? What are the two formulae used to find the area of a triangle? What is log a x - log a y equal to?

23 Divide by Π and multiply by 180 (0,1) and (1,a) (i) Plot x coords of st. points on x -axis (ii) Look at each part of f(x) separately: If rising, graph of f (x) is above x-axis and if falling, graph of f (x) is below x-axis (i) Find the midpoint of BC (M) (ii) Find the gradient of AM (iii) Use y-b=m(x-a) x = a 1 A=½base x height A=½bcsinA -sin x log a x y Show that a.b=0

24 How do you draw the graph of f(x+1) given the graph of f(x)? What is the perpendicular bisector of a line? the length of a line joining two points? How do you get the gradients of lines with equations like 3x + y = 5? What is the gradient of a vertical line? What is the Theorem of Pythagoras? What two things do you require in order to find the equation of a straight line? How do you draw the graph of f(x-1) given the graph of f(x)? What 3 processes do you go through in order to factorise a quadratic? For what values is this function undefined? f(x) (x x )(x - 3)

25 x x y 1 y1 Move the graph 1 unit to the left (i) Rearrange into the form A line which cuts the given line in half at 90 For ABC, rightangled at A, a² = b² + c² Undefined (i) Common factor (ii) Difference of two squares (iii) Trinomial The gradient of the line and a point. - and 3 Move the graph 1 unit to the right

26 How do you solve equations like cosx-5cosx=? (0 x How do you name an angle between two planes? How do you express acosx+bsinx+c in the form Rcos(x-a) etc? Given f (x) and a point on the curve, how do you find f(x)? How do you solve quadratic inequations like x - 5x + 6 0? How do you draw the graph of f(-x) given the graph of f(x)? How do you draw the graph of 3 - f(x) given the graph of f(x)? where two lines intersect? How do you draw the graph of 3 + f(x) given the graph of f(x)? the altitude AN of triangle ABC?

27 Reflect the graph in the y-axis (i) Fit in cos x-1 (ii) Factorise (iii) Solve equations Reflect the graph in the x-axis, then move it up 3 Simultaneous equations Move graph up 3 (i) Find the gradient of BC (ii) Find the gradient of AN, perpendicular to BC (iii) Use y-b=m(x-a) (i) Find lines which are at 90 to the line where the planes meet at a point (ii) Read off coordinates from angle formed (i) (ii) (iii) Expand brackets equate like terms Find R Find a b and sin tan cos (i) Integrate (ii) Fit in given point to work out value of c (i) Factorise (ii) Draw graph (iii) Read values below x-axis

28 exact values of sin(a+b), cos (A-B) etc. given that cosa=3/5 and sinb=1/13? the gradient of a line joining two points? What is sin x cos x equal to? How do you solve equations like x 0? the coordinates of a point which divides a line in a certain ratio e.g. 3:? What is a position vector? What is the sine rule? where a curve is increasing? What is the Converse of Pythagoras? the stationary points and their nature for a curve?

29 (i) Set up so you can cross-multiply (ii) Factorise and solve (i) Draw s (ii) Find missing sides (iii) Expand formula (iv) Fit in values from s AB (i) Write in the form BC (ii) Cross - multiply 3 m y y 1 (iii) Write AB (b a) (iv) Solve to find missing vector x x 1 A vector which starts at the origin tan x (i) Differentiate (ii) Let f (x)=0 (iii) Solve to find stat pts (iv) Draw nature table (v) Read values for which graph is increasing a SinA b SinB c SinC (i) Differentiate (ii) Let f (x)=0 (iii) Solve (iv) Find y-coordinates (v) Draw nature table If a² = b² + c² then ABC is right-angled at A

The Straight Line. m is undefined. Use. Show that mab

The Straight Line. m is undefined. Use. Show that mab The Straight Line What is the gradient of a horizontal line? What is the equation of a horizontal line? So the equation of the x-axis is? What is the gradient of a vertical line? What is the equation of