Year 1 - Pure Flashcards
4.5 What does the transformation of y = f(x + a) do?
Translates a by the column vector (-a,0)
4.5 What does the transformation of y = f(x) + a do?
Translates a by the column vector (0, a)
4.5 What happens to any asymptotes when a function is translated?
The asymptote is also translated
4.6 What does multiplying a constant outside a function do?
e.g. y = af(x)
Stretches y = f(x) by a scale factor a in the vertical direction (multiply the y coordinate by a)
4.6 What does multiplying a constant inside a function do?
e.g. y = f(ax)
Stretches the graph y = f(x) by a scale factor of 1/a in the horizontal direction (all x coordinates are multiplied by 1/a)
4.6 What does y = f(-x) do to the graph of y = f(x)?
Reflects the graph in the y-axis
4.6 What does y = -f(x) do to the graph of y = f(x)
Reflects the graph in the x-axis
5.1-5.4 How do you find the distance between two given points?
sqrt((Δx)^2 + (Δy)^2)
5.1-5.4 What do the equations of 2 parallel lines have in common?
They have the same gradient
5.1-5.4 How are the equations of 2 perpendicular lines different from each other?
Their gradients are negative reciprocals of each other (their product is -1)
5.1-5.4 What is the point slope formula?
y-y1=m(x-x1)
6.1 What is the midpoint of a line segment?
M((x1+x2)/2 , (y1+y2)/2)
6.2 What is the general formula for a circle which has a centre not at the origin?
(x-a)^2 + (y-b)^2 = r^2
Centre (a,b), radius r
6.3 Describe the different situations where a line intersects a circle (reference the discriminant)
2 intersections - b^2 -4ac > 0
Tangent - b^2 -4ac = 0
No intersections - b^2 -4ac < 0
6.4 What is the property of a tangent of a circle relating to the radius?
A tangent is always perpendicular to the radius at the point of intersection
6.4 What is the property of a chord relating to a circle?
The perpendicular bisector of a chord of a circle will always go through the centre of the circle
6.5 What is a circumcircle?
A circle where the vertices of a triangle are on the circumference
6.5 What is the centre of a circumcircle called and what property does it have relating to the sides of the triangle?
Circumcentre - it is the point where the perpendicular bisectors of the sides of the triangle intersect
6.5 What is the circle theorem relating to a semi circle?
The angle subtended by the diameter of the circle makes a right angle at the circumference
7.3 What is the factor theorem?
If f(p)=0 is a root of a polynomial, then (x-p) is a factor of the polynomial
OR
If (x-p) is a factor of the polynomial, then f(p)=0
7.4 What are the methods of proof?
By deduction, by exhaustion, by counter example, by contradiction
8.1 Which row of Pascal’s triangle do you use to find the coefficients for the expansion of (a+b)^n ?
(n+1)th row
9.6 Describe the features of the graph of y = sin(x)
- Repeats every 360 degrees
- Crosses x-axis at -360, -180, 0, 180, & 360 degrees
- Max. = 1, Min. = -1
- sin(-x) = -sin(x)
9.6 Describe the features of the graph of y = cos(x)
- Repeats every 360 degrees
- Crosses x-axis at -270, -90, 90, & 270 degrees
-Max. =1, Min. = -1 - cos(-x) = cos(x)
9.6 Describe the features of the graph of y = tan(x)
- Repeats every 180 degrees
-Crosses x-axis at -360, -180, 0, 180, & 360 degrees - Max. = ∞, Min. = -∞
- Asymptotes at: -270, -90, 90, & 270 degrees
10.1 What do the 4 quadrants of the CAST diagram mean?
1st quadrant - All are +ves
2nd quadrant - Only sin is +ve
3rd quadrant - Only tan is +ve
4th quadrant - Only cos is +ve
10.1 How do you find the other values for sin(x) if given the principal value?
sin(x) = sin(180-x)
OR
sin(x) = cos(90-x)
10.1 How do you find the other values for cos(x) if given the principal value?
cos(x) = cos(360-x)
10.1 How do you find the other values for tan(x) if given the principal value?
tan(x) = tan(180+x)
- Or just add/subtract multiples of 180 to x to find infinite solutions
10.2 Give all the exact trig values for sin, cos, and tan for 0, 30, 45, 60, & 90 degrees
sin0 = 0, sin30 = 1/2, sin45 = sqrt(2)/2, sin 60 = sqrt(3)/2, sin90 = 1
cos0 = 1, cos30 = sqrt(3)/2, cos45 = sqrt(2)/2, cos60 = 1/2, cos90 = 0
tan0 = 0, tan30 = 1/sqrt(3), tan45 = 1, tan60 = sqrt(3), tan90 = undefined
10.3 Give the trig identity that involves sin^2(x) and cos^2(x)
sin^2(x) + cos^2(x) ≡ 1
10.3 Give the trig identity that involves sin(x) and cos(x)
(sin(x))/(cos(x)) = tan(x)
10.4 What happens when you put the inverse of a trig function into a calculator?
It gives the principal value
10.5 Describe how you would solve: sin4x = 0
- Adjust the range to match the argument
- Draw the graph y=sinx and find the number of solutions within this range
- Manipulate to find the solutions equal to x (in this case divide by 4)
10.6 Describe how you would solve this: sin^2 (x) = 3/4
- Adjust the range to match the argument (not applicable in this case)
- Take the square root on both sides, so you have either sinx= +sqrt(3)/2 or sinx= -sqrt(3)/2
- Sketch the graph y=sinx and find the number of solutions in the range
- Manipulate the solutions to find x
11.1 How do you represent parallel vectors?
Any vector parallel to the vector a can be represented by λa where λ is a non-zero scalar
11.2 What happens when you multiply a column vector by a value?
What happens when you add two column vectors together?
λ(q, r) = (λq, λr)
(p, q) + (r, s) = (p+r, q+s)
11.3 How do you find the magnitude of a vector?
The magnitude of vector a = mod(a) = sqrt(x^2 + y^2)
11.3 How do you find the unit vector in the direction a?
a/(mod(a)), e.g. If mod(a) = 5 then a unit vector in the direction of a is a/5
11.4 What is a position vector?
A point P with coordinates (p,q) has a position vector O->P = p𝐢 + q𝐣 = (p q)
11.5 What symbols are used to show when one vector is a multiple of another?
λ and μ
12.2 How is the derivative of the curve y = f(x) written?
f′(x) or dy/dx
12.2 What is the formula for calculating the derivative of a function using first principles?
f′(x) = (lim(h->0))((f(x+h) - f(x))/h)
12.3 How do you differentiate y = x^n or f(x) = x^n?
y = x^n, dy/dx = nx^(n-1)
OR
f(x) = x^n, f′(x) = nx^(n-1)
12.4 How would you differentiate the a quadratic with equation y = ax^2 + bx + c?
y = ax^2 + bx + c, dy/dx = 2ax + b
12.5 How do you differentiate a function with two or more terms?
Differentiate the terms one at a time
12.6 How do you find the tangent to a curve when give an x value and the equation of the curve? How do you find the normal to a curve when given an x value and the equation of the curve?
Tangent:
1. Find the derivative of the equation of the curve
2. Substitute the x value into the derivative to find the gradient at the tangent
3. Substitute the x value back into the equation of the curve to find the y value at the tangent
4. Use y-y1=m(x-x1) to find the equation of the tangent
Normal:
Same steps except use the negative reciprocal of the gradient of the tangent
12.7 What does it mean when a function f(x) is increasing in an interval?
f′(x) ≥ 0 for all values of x within the interval
12.7 What does it mean when a function f(x) is decreasing in an interval?
f′(x) ≤ 0 for all values of x within the interval
12.8 What is a second order derivative? What is the notation?
Differentiating a function twice gives the second over derivative
f(x) -> f′(x) -> f′′(x)
y -> (dy)/(dx) -> (d^2y)/(dx^2)
12.9 What is a stationary point on a curve?
Any point on the curve y=f(x) where f′(x) = 0
12.9 What are the three types of stationary points? How are they different?
Local minimum, local maximum, point of inflection
Loc. min. - -ve grad. to +ve grad.
Loc. max. - +ve grad. to -ve grad.
POI - either +ve grad. to +ve grad. OR -ve grad. to -ve grad.
12.9 If a function f(x) has a stationary point at x=a, how do you what kind of stationary point it is?
If f′′(x) > 0, then the point is a local minimum
If f′′(x) < 0, then the point is a local maximum
If f′′(x) = 0, then it could be a local minimum, local maximum, or a point of inflection. You need to look at the gradient of the points either side of the stationary point to figure out what type of stationary point it is.
12.10 How do you sketch the gradient function of a function y= f(x)?
Gradient function is the graph of y = f′(x), therefore find the derivative of f(x) and sketch that
OR
Consider the gradient at the stationary points (zero) and these will be the roots of the gradient function, then consider what the gradient is doing (+ve/-ve, increasing/decreasing) at different sections of the graph
12.11 What is a way to represent the change in volume, V, over time, t?
dV/dt
