## Saturday, 20 April 2013

### Coordinate Geometry - about a SLOPE

 The Slope of A Line source: http://math.about.com Identify a few concepts on (i) gradient properties (ii) equation of a line (iii) trigonometry and gradient and(iv) collinearity from the article below. Post as comment. When the slope of the line is 0, you know that the line is horizontal and you know it's a vertical line when the slope of a line is undefined. In the Figures below, the subscripts on point A, B and C indicate the fact that there are three points on the line. The change in y whether up or down is divided by the change in x going to the right, this is the 'rise over run' concept. y = mx + b is the equation that represents the line and the slope of the line with respect to the x-axis which is given by tan q = m. This is the slope-intercept form of the equation of a line. (m for slope? Seems to be the standard!) When the slope passes through a point A(x1, y1) then y1 = mx1 + b or with subtraction  y - y1 = m (x - x1) You now have the slope-point form of the equation of a line. You can also express the slope of a line with the coordinates of points on the line. For instance, in the above figure, A(x, y) and B(s, y) are on the line y= mx + b : m = tan q =  therefore, you can use the following for the equation of the line AB: The equations of lines with slope 2 through the points would be: For (-2,1) the equation would be: 2x - y + 5 = 0. For (-1, -1) the equation would be: 2x - y + 1 = 0

1. i) The gradient can be positive (going upwards) or negative (going downwards). It is commonly defined as how steep/gentle the slope is.
ii) The equation of a line (assume it's linear) is y = mx + c. 'm' refers to the gradient of that select line, while 'c' refers to the y-intercept of the line.
iii) In trigonometry, two triangle sides have a gradient about a certain base. If we do not have the base to compare it with (base gradient = 0) we would be unable to have no specific gradient.
However in coordinate geometry we have the x-axis to be m=0, thus all sides of the triangle would have a gradient.

iv) The idea of collinearity comes in when we want to calculate gradient. We identify two points that are on the line we wish to calculate the gradient of, in this case both points are collinear. Collinearity refers to both ordered pairs resting on the same line.

2. (i) gradient properties: if the graph is a line then it will form an angle with x-axis , if the graph is a curve then the tangent of a certain point will form an angle with x-axis, assume the angle is alpha, then tangent alpha is the gradient of this line (or of this point). If the gradient is zero then it is horizontal, if the gradient is undefined then the graph is a vertical line.
(ii) equation of a line : y=mx+b, m is the gradient mentioned above, b will determine the interception with y-axis, two points will determine one line so only need the coordinates of two different points then can know the expression of the line.
(iii) trigonometry and gradient : like I mentioned before, tangent alpha is the gradient.
(iv) collinearity from the article below:three or more points with the same gradient and have an area of zero will be on the same line.

3. 1. A gradient is a slope which can be calculated using the 'rise over run' method. a gradient of 0 would mean that the line would be straight.

2.The equation of the line has a format of y = mx + c, where m = the gradient of the slope and c is the intersection with the y-axis.

3.The tangent of the line would be the gradient as the gradient of the line is rise over run and the tangent of the line would also be the rise over run.

4. The area of ABC is 0 and the gradient of the three points in the graph of the article are the same, therefore they are collinear.

- Positive if line is going upwards , negative if otherwise, 0 if straight line
- Often found with rise/run

2. Equation of a line - y = mx + b
b being the y-intercept

3. Trigo can be used to find the gradient
Trigo - tan"ANGLE" = rise/run
Angle is where the line makes the angle with the x-intercept.

4. Collinearity
- Can be found between points using 2 methods
1. Finding the gradient for all 3 points
2. Find the area of the "shape" made from the 3 points

5. i) Gradient can be positive and negative, the further the gradient is from 0 the steeper it is
ii) y=mx+c
iii) Gradient = rise/run = tangent
iv) Collinearity i when there are 2 ordered points on the same line.

6. (i) For a line with a positive gradient, the line will be sloping upwards. For a line with a negative gradient, the line will be sloping downwards. For a vertical line, the gradient is undefined. For a horizontal line, the gradient is 0.

(ii) The general form of the equation of a line is y = mx + c where m is the gradient of the line and c is the y intercept of the line

(iii) Gradient is rise/run, so the gradient of a line can also be calculated by using tangent of the angle formed by the line and a horizontal line, since tangent is also rise/run

(iv) For points to be colinear, they must all be lying on the same line. This means that the must have the same gradient and the area between these points is 0

7. 1. the slope of the line
2.represents the line and the slope of the line
3.x1-x2 and y1-y2(if negative both take modulus) are the length of two right-angle sides of one triangle respectively and the gradient is the length of the hypotenuse.
4.when every line the two points in the three formed has the same gradient, the three points are collinearity.

1. 2. expression y=mx+c

8. Identify a few concepts on
(i) The steepness of a slope is dependent on this. It is negative if it points to the south direction, it is positive if it points to the north direction
(ii) y=mx+c for linear, where m is the gradient and c is the y-intercept.
(iii) The gradient is represented by tan q, where q is the angle of elevation of the line from the x axis
(iv) Collinearity is the calculation of gradient using two ordered pairs, where both rest s on the same line.

9. i) If the gradient is positive , the graph will be going upwards towards the right , if it is negative however , the graph will be going down wards towards the right . A general shape of a positive graph will be "/". A general shape of a negative gradient graph is "\" Besides the signs , another factor that affects gradient is the coefficient of x .

ii) The equation of the line is the 'relationship of the line' . If we know either the x or y value and we put it in the equation , the other term will correspond on the line . All colinear points satisfy the sane equation . It has the general form , y=mx+c . M is the gradient of the graph and c is the y-intercept .

iii) Gradient is also known as the tangent . It can be used to determine the gradient of a point on another graph . Tangent is also the function Opposite / Hypotenuse that trigonometry commonly uses

iv) They must satisfy the equation and the gradient is similar for all points .

10. (i) A horizontal line has a gradient of 0, while a vertical line has undefined gradient, the steeper the graph (more change in y-axis over 1 unit of x-axis) the higher the gradient, also negative gradient means the line is in the other 2 quadrants

(ii) The equation of a linear line is y = mx + c, m is the gradient and c is the y-intercept of the line. (The point where the line cuts the y-axis)

(iii) Gradient = rise / run, however is trigonometry, it also means the opposite over the adjacent if we take the angle between the line and the x-axis, thus tangent = gradient

(iv) Collinearity means that the points are all resting on the same point and therefore have the same gradient

11. (i) The gradient of a line can be both positive and negative. A positive gradient goes upwards and a negative gradient goes downwards. It can be steep or gentle

(ii) The equation of a line is y=mx+c, where 'm' is the gradient and 'c' is the y-intercept.

(iii) Triangles are made up of lines which has gradients. The gradient of a line is it's rise/run, which is also the tangent.

(iv) It is a set of points lying on a line. It is the approximate linear relationship between two values.

12. i) The gradient can be positive (Slope Upwards), negative (Slope Downwards, zero (Horizontal Line) or undefined (Vertical Line).
ii)f y = mx + c Where m is the gradient and c is the y intercept.
iii) Gradient is the rise over run, so it can also be calculated by using tangent of the angle which is the opposite over the adjacent.
iv) The points must be on the sane line

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Gradient is often defines as the steepness of the slope and is calculated by the Rise over Run. If the gradient is positive, the line would go up; if it is negative it would would down. If the line is horizontal, the gradient is 0 and if it is vertical, the gradient is undefined.

(ii) equation of a line
The general form of a line is y=mx+c where m refers to the gradient of the line, and c refers to the constant, often which is the y-intercept.

Tangent can be found be taking the opposite side divide by the adjacent side of a right angle triangle. Moreover, the opposite and adjacent side can also be interpreted as the Rise and Run, thus making the two terms similar.

(iv) collinearity from the article below
It means that every point on the same line has the same property eg. gradient between points.

15. (i) The gradient can be positive or negative and is defined by rise over run. If the gradient is 0, it is a horizontal line. If the gradient is undefined, it is a vertical line.
(ii) An equation of a line is represented by y=mx + c. m is the gradient while c is the y intercept.
(iii) Tangent is the rise over run hence we can use the tangent to figure this out/
(iv) Collinearity is a property of a set of points, specifically, the property of lying on a single line.

16. 1. positive if the graph go upwards to the right; negative if it go upwards to the left

2.y= mx+c (c is the y-intercept); m is the gradient of the graph

3. Tangent of the graph is the gradient

4. Collinearity is used to find the equation of the graph.

17. (i) The gradient of a line can be positive, negative, 0 or undefined.
For the gradient of the line to be undefined, it will have to be vertically upwards.
If the gradient of the line is 0, the line would be horizontal.
If the gradient of the line is positive, the line would be going horizontally upwards.
If the gradient of the line is negative, the line would be going horizontally downwards.

(ii) The equation of the line is y=mx+c. m being the gradient and c being the y-intercept.

(iii) Gradient of the line is rise/run. As for the trigonometry, Tangent would be opposite/adjacent about an angle in a right-angled triangle. Hence, tangent would be the gradient.

(iv) Collinearity means that all of the points are lying on the same line with the same gradient.