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On a graph, with the line, Y=n^x, if a number has too many decimal places then you may round it.
With the graph function: y=f(x)±a, the line on the graph moves along the y axis by the value of ±a.
The graph function that leads to the line(s) on a graph moving up or down the y axis by the value of 'a' is written as y= f(x)±a.
If y=f(x+ a), then there's a change on the values on the graph on the 'x' axis by the value of -a. If y=f(x-a), then there's a change on the values on the graph on the 'x' axis by the value of a.
The function that leads to the line(s) on a graph moving right or left along the 'x' axis is written as y= f(x±a). If that function is applied to a y='x'² line(where y=f(x)), then the equation of the line (y='x'²) changes to y=(x±a)².
If y=kf(x), then the values of the points for the y-axis are multiplied by the value of k, so if k=0.5, they are halved. If y=f(x)=x², when put into this functional transformation, it turns to y= kx².
If y=f(kx), there is a multiplication of the x-value inside the function by k. Where y=f(x)=x², the transformation leads to y= (kx)². If y=x²-bx+c, the transformation leads to y= (kx)²-b(kx)+c. Additionally, the x axis value of the points on the graph are multiplied by 1/k.
If the line: y=x² is put through the function; y=f(k'x'), the equation of the line changes to y=(kx)².
If the line y='x'² is put through the function; y=kf('x'), the equation of the line changes to y=kx².
The vector for y=f(x)±a is (0,±a), while the vector for y=f('x'±a) is (±a,0).
In alphabetical order, the 4 types of transformation are enlargement, reflection, rotation and translation.
With 'reflection', the mirror line is known as a line of reflection.
If a shape is reflected by an axis of reflection, from the 'axis of reflection', both shapes are equally apart, but compared to the other, they are backwards.
If a shape is reflected along the line: y = x, then, for the reflection, the values of y and 'x' from the points on the original shape swap.
To describe a rotation, the three pieces of information used are the angle of rotation, the direction of the rotation and the centre of the rotation. The direction of rotation can be either clockwise or anticlockwise. The center of rotation is the point about which the shape is rotated.
When shapes are being enlarged on a graph, the point from which enlargement takes place is called the centre of enlargement.
With 'enlargement', the value a shape is multiplied by, is known as the scale factor.
With a negative scale factor, in terms of the centre of the enlargement, the new shape will appear on the other side (of the centre of enlargement).
To find the 'centre of enlargement' after drawing two shapes, draw lines that are straight which connect corresponding corners of both the first-drawn and second-drawn shapes. The point where all the lines meet is the centre of enlargement.
If shapes are 'similar' then corresponding angles are equal and corresponding lengths are in the same ratio. Using the ratio multiply or divide to find unknown lengths.
While working with similar shapes if the lengths are in the ratio; a:n, the areas are in ratio: a²:n². If areas are in the ratio; a²:n², the volumes are in the ratio: a^3:n^3.
In a triangle in a semicircle (all corners touching the circumference), the angle subtended by the diameter is equal to 90°.
Angles in the same segment (angles at the circumference, subtended by the same arc) are equal.
The angle subtended at the centre of a circle, subtended by an arc, is twice any angle subtended at any point on the circumference.
With a circle, a cyclic quadilateral is 4-sided shape with each vertex touching the circumference of the circle. Opposite angles in a cyclic quadrilateral (in terms of the circle) add up to 180 degrees. The exterior angle of A is equal to the angle that's opposite A.
With a circle, the lengths of 2 tangents from a point are equal.
If some-thing subtends, then it extends underneath. In circles, angles subtended by the same arc or chord are equal.
If a chord is drawn in a circle, it splits the circle into 2 segments; the larger one is known as the [] [], while the smaller one is called the [] [] major segment--minor segment--. The angle made by the point of contact of a chord and a tangent, and the angle that is in the alternate (i.e. other) segment, subtended by the same chord, are equal.
The limits of y = sine θ¸ are both 1 and -1. When y='sine x', the line on the graph looks like a w.. ..ave that repeatedly goes above and below the 'x' axis every(in degrees) 180 degrees or (in radians) pi radians.
A right angled triangle has 'c' the hypotenuse, 'b' the horizontal leg and 'a' the other side. In terms of sides, a/c= sine('theta') {which in terms of csc('theta') is csc('theta')= 1/sine(theta) and in terms of sides; csc('theta')= c/a.
A right angled triangle has 'c' the hypotenuse, 'b' the horizontal leg and 'a' the other side. In terms of sides, b/c= cos('theta') {which in terms of sec('theta') is cos('theta')= 1 / sec(theta) which in terms of sides equals c/b.
'A right angled triangle has 'c' the hypotenuse, 'b' horizontal and 'a' vertical'. In terms of sides, a/b= tan(theta), which in terms of just trigonometric identities is tan(theta)= sin(theta) / cos(theta). {1/ tan(theta)= cot(theta) or in terms of sides= b/a.
The Sine Function (y=sine(x)) has a line shaped like an up-down curve, which repeats every (in radian) 2pi radians, or (in degrees) 360°.
The width of the repeating pattern that is measured on the horizontal axis, is called the period and is measured in radians. One π radian in degrees is 180°. Angle in degrees = angle in radians x []/[] 180°/π.
The formula to calculate the length of a circular arc is 'θ'/360° x 2πr.
The area of a triangle with 2 sides as well as 1 angle labelled, with 'a' a length, 'b' a length and 'c' the included angle can be given as 0.5ab x sine c.
The formula to find the area of a sector is 'θ'/360° x πr², which expresses the sector as a fraction of the area of a circle.
To work out the area of a segment, first calculate the area of both the sector and triangle (within the sector), then subtract the area of the triangle from the area of the sector.
The expression for the volume of a sphere is 4/3πr^3 and the formula for the surface area is 4Ï€r².
When 'h' stands for height and 'b' stands for the area of the base, the volume of a pyramid is bh/3, while the volume of a cone is Ï€r²h/3.
If 'b' stands for base area, 'l' stands for lateral area, 's' stands for slant length and 'p' is the perimeter, then the surface area of a pyramid with side faces the same is (in terms of p,s&b) 0.5ps+b, but if the side faces aren't the same then the expression is b+l.
{A pyramid is made by joining a base to an apex.
{A object shaped like a cone is known to be conical.
{In a way, a cone can be made by spinning a triangle.
If a cone shares the same base area and height with a cylinder, then the volume of the cone is equal to the volume of the cylinder divided by 3.
If 's' stands for side length, the surface area of the curved side of a cone is either πrs or πr√(r²+h²). So to find the total surface area, add to either of them πr².
If 'a', 'b' and 'c' are lengths, then a formula with abc (or a^3, a²b etc) refers to volume. Formulae such as a²b+bc have no dimensions because things like volumes, areas and lengths don't add together.
Vectors a and b are parallel if there exists a real number c such that a=cb. All values on a number line including all values greater than -∞ and smaller than ∞ are real numbers. √-1, ∞ an -∞ are not real numbers.
{If 'w' represents the width and 'l', the length, then the area of an oval is equal to 0.8wl.
{Where 'a' is half of the major axis and 'b' half of the minor axis, using π instead of 2π, the circumference of an oval(c)≈ π(3a+3b-√(3a+b)(3b+a)).
{The technical term for an oval is an elipse.
{In a kite (ABCD) if the length of AC=x, and BD=y, then the area of the kite is xy/2. Where 'a' and 'b' are the lengths of two unequal sides and C is the angle between them, the formula for the area of a kite is ab x sine C.
Notations like (c,d) (usually 1 on top of the other), a(underbar), AB (with an arrow above), a(by an arrow) and 'a' in bold are ways of writing vectors.
The direction of a vector is usually shown by an arrow.
If it's a right-angled triangle that's being used, then the magnitude of the vector may be found using 'Pythagoras' theorem'.
The resultant of multiple vectors can be found by adding them. The resultant may be labelled with 2 arrowheads.
When adding or subtracting vectors that are in a 'column' form: add or subtract both the top and bottom parts seperately.
With vectors: a-b is equivalent to a+ -b.
With vectors: if c=(3,4), then -c= (-3,-4)and if c=(-3,4) then -c= (3,-4).
Scatter diagrams are used to show correlation.
With Scatter diagrams, if the values of both 'subjects' are increasing, then there is a positive correlation. With positive correlation on a scatter graph, the plots 'rise' to the right, but with negative correlation, the plots 'rise' to the left.
If the plots on a scatter graph aren't reasonably close together or show little uniform direction, then there is either little or no correlation.
∑ means sum(of).
Data that can only take certain values, like the amount of students in a class(you don't get half a student) is known as discrete data. Data that can take in any value within a range (like times in a race which could be measured to fractions of a second) is known as continuous data.
With continuous data: when the data is grouped into class intervals, the exact data is not known.
Moving averages can be used to smooth out changes in a set of data over a period of time.
The formula to find the mean of continuous/grouped data is: []=[] x(over bar)= ∑fx/∑f, where x(over bar) represents the mean, 'f' represents the frequency and 'fx' can be found by multiplying both the frequency and the midpoints of the intervals.
The position of the median of grouped data is given as Σf+1/2, where ӈ‘f is the sum of the classes' frequencies.
To find the n-point moving average find the average of the first n numbers. Repeat through all the data values, with position of the number you start with, each time increasing by one until you have included the last data value. You may then make a list of the discovered averages. The averages may then be plotted onto a graph and roughly through the points as a sort of line of best fit line, you may draw a straight trend line.
With a scatter graph, instead of drawing an estimated line of best fit, it may be better to find the mean.
On cumulative frequency graphs, if 'n' is the value on the y-axis at the pinnacle of the cumulative frequency curve, then the formula for the upper quartile is: 3(n+1)/4, the median is: (n+1)/2 and the lower quartile is: (n+1)/4. From these points on the y-axis, a straight point may be drawn side-ways to the ogive, then down-wards to the values of the quartiles and median on the x-axis.
With continuous data, the class intervals can be written with inequalities(using a, b and c in consecutive order as a≤b<c (where b may be an abbreviation of the unit being measured or counted).
On a graph: the inter-quartile range is the distance between the upper quartile and lower quartile. The range can be found by subtracting the lower quartile from the upper quartile.
A large inter-quartile range indicates that the 'middle part of the data is widely spread around the median. A small inter-quartile range in concentrated about the median.
A box plot is sometimes known as a 'box and whisker plot'.
A graph with a box, 2 lines from each side that may sometimes be called 'whiskers' and a scale underneath is called a box plot.
On each side of a box plot there is a line, sometimes called 'whiskers' and underneath it there is a scale.
In a box plot, the far end of the line drawn to the box from the left represents the lowest value, while the far end of the one from the right represents the highest value. Of the box: the edge to the left represents the lower quartile, the edge to the right represents the upper quartile and the line in the middle represents the median.
In a box plot, the width of the box represents the inter-quartile range.
The current total of all the frequencies is known as the cumulative frequency.
Histograms are similar to bar charts except the bars can be different widths. The area of each bar represents the frequency.
When drawing histograms, the vertical axis is often(even when not labeled): frequency density.
When drawing histograms, the frequency that the rectangles represent is equal to their a[] area. The formula for the frequencies is frequency= frequency density x class width and frequency density(which can be found through 'rearranging formulae') = frequency/ class width.
If events 'a' and 'b' can't simultaneously occur then the events are mutually exclusive. In that case: p(a or b)= p(a)+p(b).
Independent events are events that aren't affected by previous events. The probability of multiple independent events can be found by multiplying the probabilities together. P(a and b and c...)= P(a) x P(b) x P(c)...
Vectors are equal if they have both the same magnitude and direction. The inverse of a vector has the same magnitude but has the opposite direction. The inverse of the vector AB(arrow overhead) is (without adding any additional characters or symbols): the vector BA(arrow overhead). With that rule excluded the inverse of the vector AB(arrow overhead) can be -AB(arrow overhead).
To add 2 vector is to apply the first, then apply the second. CD(arrow overhead) + DE(arrow overhead)=(as a vector:) CE or simply E= c+d. This is known as the triangle law.
Multiple probabilities add up to 1.
With tree diagrams, the 'branches' are often labelled with probabilities and the sum of all probabilites from the same source is 1. At the end of the branches, there are the outcomes.
An dependent event is an event affected by previous events.
With dependent events, one or both of the numerator and denominator of the probability may change depending on previous outcomes.
When 's' is side, 'a': angle, 'r': right-angle and 'h': hypotenuse, then 4 thing that makes a triangle congruent are if they have the same(using the letters given): 'sas, sss, aas, asa or rhs'. In the triangle, they should occur in the given order (whether [] or anti[]) clockwise--clockwise--.
To prove the sum of angles in triangle 'ABC'(labelled left to right): draw a straight line(OP) through 'B'. Angle 'OBA' and angle 'OBC' are equal(respectively) to angle 'A'(CAB) and In thangle C(ACB) because they are alternate angles . The third angle in both the straight line and the triangle is B, thus there're an equal amount of degrees in both of them, with that amount being (because of the number of degrees in a straight line) 180°.
The sum of angles in a quadrilateral resulting in 360° can be proved by there being in a quadrilateral: 2 triangles as the sum of angles in a triangle is 180°: the sum of angles in a quadrilateral is 180° x 2.
A square diagonally bisects at its vertices.
In triangle 'ABC', with each side (written in lowercase) opposite its namesake angle: the sine rule is either a/sine(A) = b/sin(B) = c/sin(C) or sin(A)/a = sine(B)/b = sin(C)/c.
To find an angle or length using the sine or cosine rule: find the right formula, then substitute the letters for values before (if 'needed') 'working out' some values.
A triangles area is 0.5ab x sine C, where C is located between 'a' and 'b'.
In a sine graph, where y=f(x)=sin(x): y=k*sine(x) affects both a[] amplitudes by multiplying them by k.
In a cosine graph, where y=f(x)=cos(x): y=k*cos(x) affects both a[] amplitudes by multiplying both by k.
In a circle: there are 2 triangle, next to each other, with the joining of both of the buttom edges making a diameter and both of them having 2 corners touching a corner from the other triangle (once at the origin, and the other at the []'s circumference) trianglecircle--. The angles not at the origin in the left triangle are 'a' and all the angles not at the origin in the right triangle are 'b' (the triangles are isoceles as two of their sides are radii). To proof that the angle at the circumference, subtended by the diameter is 90°: in the larger triangle (made from the smaller two): 180° = a+a+b+b or with brackets: 180° = 2(a+b). Thus a+b = 90°.
The perpendicular bisector of a chord passes through the center of the circle and is the diameter.
An angle at the circumference of a circle and subtended by the diameter, which is in a triangle that has two of its sides touching the diameter (the third side), equals 90°.
The hypothetical trapezium has top side: a, buttom side: b and length between: h. To prove the area of the trapezium: draw an adjacent and connected trapezium - of the same dimensions (with the same labelling) but with the difference that it is inverted. The new shape is now a parallelogram. The formula for the area of a parallelogram is vertical height x base length (h * a+b), but as this is two trapezia, the formula for one trapezium is (in terms of the formerly stated letters) h(a+b)/2
G/cm^3, g/m^3, kg/cm^3 and kg/m^3 are some ways to display density. Density= mass/volume.
When sampling for handling data, in attribute sampling, the selection of the sample is made by choosing an attribute, which is [] to the variable being investigated.
If 'c' is a vector, kc is the vector 'c' x k.
At times: the magnitude of a vector is shown with, on either side, either 1 or 2 vertical bars. It may be shown with two vertical bars on each side so as to not confuse it with the absolute value.
When using vectors: 'ordinary' numbers (such as 7, -0.78, 156) are known as scalars.
Vectors can be written in brackets with one character being on top of the other or both being next to each other. In this case: the left value or the one on top represents the movement horizontally, while the other represents the movement vertically, but that is on a 2-d graph. On a 3-d graph the first value represents movement like on the x-axis, the second value represents movement like on the y-axis and a third value is used which represents movement like on the z-axis.
If the vector d=(a,c), the magnitude can be calculated, if in a right-angle triangle, using Pythagoras' theorem with, in this case, the formula: d= √a²+c².
The quantities that you multiply a vector by are called scalars. Doing that is called scaling.
AB+BC(arrows overhead) = AC(arrow overhead). If vectors d+e=f and vectors d+c=f, then both sets and their routes are equal.
Subtracting a vector is the same as adding that vector's inverse.
Scalars have magnitude but not direction.
{You mark a point by how far along and how far up it is (x,y) in the coordinates system known as Cartesian coordinates. A point is marked by how far away it is from the origin, and what angle is between the line connecting it to the origin and the x-axis (r,θ) in the coordinates system known as Polar coordinates.
{To covert from Cartesian to Polar coordinates:r = √ ( x² + y² ), thus making it a hypotenuse and 'θ'= tan^-1 ( y / x ). To convert from Polar to Cartesian coordinates: x = r × cos( θ ) and y = r × sin( θ ).
{If the four quadrants of a graph are numbered anticlockwise 1-4 starting at the top right: when converting from Cartesian to Polar coordinates: if the point is in quadrant 1, use the calculator value given for θ = tan^-1(y/x), but if it is quadrant 2 180° and if the point is in the fourth quadrant, add to the given value: 360°.
Sine^-1= arcsine, cos^-1= Arccos and tan^-1= arctan.
{The centroid of a triangle is the point of intersection of its medians (the lines joining together each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1, which is to say it is located ⅓ of the distance that is upwards from the midpoint of the buttom side to the top angle.
To flip graph f(x) upside down, the function transformation used is -f(x). Any point on the x-axis, after transformation would be on the x-axis; the points off the x-axis change side. When applied to the graph y=x²=f(x), it changes to y= -x².
To flip graph f(x) left-side right, 'depth rotating' it around the y-axis: the function transformation applied to f(x) is f(-x). Any point on the y-axis, after transformation, would be on the y-axis. The points off the y-axis change side. The mirror image of g(x), as reflected in the y-axis is g(-x), when this is applied into the line: y=g(x)=x^3: the equation of the line changes to (with brackets) y= (-x)^3.
Where y=cos(x+k) or y=sine(x+k): there is a change in the points on the graph in the values on the x-axis by a value of -k.
y= cos(kx) There is a change in a direction that's horizontal, as horizontally the graph is squeezed by a factor of k. Alternatively, the x axis value of the points on the graph are multiplied by 1/k.
A law which states that if a body is acted upon by two vectors represented by two sides of a triangle taken in order, the resultant vector is represented by the third side of the triangle is known as the triangle law.
The cosine rule forumla for finding sides, where the known sides amount to 2 and there's an angle which is [] them between them is a²=b²+c²-2bc x cos(A) or b²= a²+c²-2ac x cos(B) or c²= a²+b²-2ab x cos (C).
The cosine rule formula to find an angle where the known lengths of sides amount to 3 is cos (A)= b²+c²-a²/2bc, which was found through rearranging formulae.
The sine rule is used when you know both [] side and [] an[] one side and two angles, or the amount of sides known amount to 2 and opposite one of the 2 sides is a known angle.
A circle has two segments: angles a and b are in different segments, so they are said to be in alternate segments.
If there is a right-angled triangle with θ between the hypotenuse and the adjacent, in terms of the sides: sineθ = opposite/hypotenuse; cosθ = adjacent/hypotenuse; tanθ = opposite/adjacent.
{The notation fg means carry out first function g, then function f. Sometimes, fg is written as fog. The function which reverses another function is called and inverse function. For example the inverse of y = 2x is x = y/2.
The vector law that shows that going from A to C via B is the same as going from A to C via D is known as the parallelogram law.
Given a quadratic function ax^2+bx+c, the vertex is where x= -b/2a.
sin(x) = cos(x- 90°)
If the median is in a position n.5, take it to be the mean of the values in positions n and n+1.
The graph of y = sin θ start at the o[] origin. The reaches its maximum amplitude - 1, where x (or θ) = 90°. It then goes to its minimum amplitude of -1, where x = 270°, before going back to the x-axis, where x = 360°. The process continues to repeat.
The period of the tan function is 180° and every 180°, the value of the function is infinite, starting at 90°.
The data handling cycle: (1)Specify the p[] and p[] problem and plan, (2)collect data from a variety of sources, (3)p[] and r[] data process and represent data, (5)[] results evaluate--.
When trying to solve graphical inequalities graphically and there isn't (>, ≥, < or ≤)0, you can make it like that through rearrangement.
When solving a quadratic inequality graphically, with an inequality that includes -x²: the graph is like the graph of an equation that includes x², but inverse (upside down).
If the equation of a graph can, through factorisation, turn to (x+n)(-x+o), then (-x+o)) can be changed to (o -x). The roots of the graph (with equation: (x+n)(o-x) are -n and o.
To solve a simultaneous equation: 1)rearrange the equations into the form(where 'a', 'b' and 'c' are (possibly negative) numbers): ax+by=c, . 2) Match up the coefficients of either the x's or y's, by multiplying 1 or more of the equations by a suitable number. 3) Find the difference between both equations and divide it by the difference between the non-matching cooefficient to find the value of one of the variables, which can be used to find the value of the other variable.
Proof that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference. Construct the situation: mark two points that are below the [] and that are on the [] centre-- circumference, then mark another point that is above the [] and that is on the [] centre--circumference-- (for the angle 'b'). Join each point below the centre to both the point at the circumference and the centre (making angle 'a'). [#A25]. Add a radius from the centre to b, making two triangles that are isoceles due to the radii. Label: label the equal angles in the triangle to the left 'w' and the angle at the origin 'x', then label the equal angles in the triangle to the right 'y' and the angle at the origin 'z' [#A26]. Using the angles in their triangles: x=[] and z=[] x=180°-2w and z=180°-2y. Using those values of x and z, and the sum of angles around a point: 360° = a+x+z =[...], which simplies to 0 =[...] 360°=a+(180°-2w)+(180°-2y), which simplifies to 0 = a-2w-2y, therefore (with brackets) a = [], thus a = []b a = 2(w+y), thus a = 2b. Q.E.D.
Proof that angles subtended by the same arc, and are at the circumference are equal. Construct the situation: mark the circle's [] and draw a c[] centre and draw a chord. Draw two triangles in the same segment that share the [] as one of their sides chord--. In each triangle: label the angle opposite the chord. Label the angle at the left: a and label the other angle: b [#A27]. Using the points where the chord touches the circumference as vertices: draw a triangle with the other vertex at the centre of the circle. Label the angle at the centre - c [#A28#]. c = []a = []b c = 2a = 2b (based on the circle theorem that states: the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference). c = 2a = 2b, therefore 2a=2b, thus a = b. Q.E.D
To proof the sum of opposite angles of a cyclic quadrilateral(in relation to the circle) is 180°. In the circle: mark the centre. Create a cyclic quadrilateral, with the vertices at the circumference. Label the two opposite angles (a and b). To each of the other vertices: draw a radius. As an angle subtended at the circumference by an arc is half that subtended at the centre: the angles at the origin are 2a and 2b [#A51#]. 2a + 2b = 360° (which is the sum of angles around a point). Therefore a + b = 180°. Q.E.D.
[#A24]We want to prove the alternate angle theorem that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. Construct an alternate segment situation in a semicircle. Remember to add a tangent to the diameter enclosing the semicircle. Label as shown in [#A24]. When tangents meet with radii: an angle is formed of 90°. Thus BCA + z = 90°. As it is subtended at the circumference by the diameter, angle ABC = 90°. ABC + BCA + y = 180° (due to the sum of interior angles in a triangle). Simplifying the equation gives BCA + y = 90°. We have established two addition equations that result in 90°: BCA + y = 90° and BCA + z = 90°. You can state that in order for them to be true: [] = [] z = y. Q.E.D. (That can be shown through more rearrangement.)
With transformation, with rotation, before and after rotation, each point remains exactly the same distance from the [...] centre of rotation.
With transformatin, with enlargement, if the scale factor is s, each point will become further away from the centre of enlargement by a factor of s.
The surface area of a cylinder equals 2πr² + 2πrh = 2(πr² +πrh), whereas the surface are of a cone is πr² + πrs.
When working with continuous data, in the first column, the intervals are arranged in order of size from smallest to biggest. The other 3 columns are 'f' 'x' and 'fx'. 'f' stands for frequency, 'x' represents the midpoints of the interval and 'fx' can be found by multiplying the frequency and the mid-point. The '∑' of 'f' and 'fx' can be found on the row at the bottom.
To draw a cumulative frequency graph from a table of data, plot points, with their x-value being the [] value of each [] interval top--class--, and their y-vaue being the cumulative frequency. Afterwards [] the points with a smooth [] join--curve--. The line starts at the earliest point.
Created by: Toluo