Introduction To Graphs - Class 8 Maths - Extra Questions

Coordinates of the point of intersection os X- axis and y-axis is $$(0,0)$$ i.e. origin

The process of fixing a point with the help of the coordinates is known as plotting of the point.

The y-coordinate of the point $$(2, 4)$$ is $$4$$.

Abscissa or the $$x$$-coordinate of a point is its distance from the $$y$$-axis.

A point is denoted by $$ (x-$$cordinate, $$y -$$ cordinate$$) $$

The horizontal axis in the coordinate plane is called the $$x-$$axis. The vertical axis is called the $$y-$$axis. The point at which the two axes intersect is called the origin. The origin is at $$(0,0)$$ on the $$x-$$axis and $$0$$ on the $$y-$$axis.

A rotation $$90^o$$ clockwise about the point (2,4) does not result in a diagram that has line symmetry or rotational symmetry.
The diagram does not have line symmetry because there is no line on which a mirror can be placed so that the polygon is the reflection of the other. The diagram does not have rotational symmetry because there is no point about which it can be rotated so that it coincides with itself. 

A point is denoted by $$ (x-$$cordinate, $$y -$$ cordinate$$) $$

From the above given graphs, it is clear that the red graph is a bar graph as the bars are not joined and are separate.While the orange graphs, the bars are all together. 

The line which passes through $$(3,2)$$ and $$(2,3)$$ cuts the $$x-$$axis at $$(5,0)$$ and $$y-$$axis at $$(0,5)$$.

Abscissa is $$x$$ co ordinate

Since centre is the mid point of the end points of the diameter
Let the other end of diameter is $$(x, y)$$

A histogram is used to graphically summarize and display the distribution of a process data set.Creating a histogram provides a visual representation of data distribution. Histograms can display a large amount of data and the frequency. The function will calculate and return a frequency distribution. 

The $$x$$-coordinate is called as abscissa.

We have, $$(-2,4)$$ is the point lying on the second quadrant, $$(3,-1)$$ is the point lying on the fourth quadrant, $$(-1,0)$$ is a point lying on the negative half of the $$x-$$ axis, $$(1,2)$$ is the point lying on the first quadrant, $$(-3,-5)$$ is the point lying on the third quadrant.

Given, $$(x+y,1)$$ and $$(3,x-y)$$ be equal ordered pairs.

For any point $$A(x,y)$$, reflection in $$x-axis$$ is given by $$A'(x,-y)$$.

Bar graph has gaps between the bars, as seen in the "Categories" graph.

To describe the position of a table lamp on the study table, we have two take two lines, a perpendicular and horizontal. 

$$A'$$ is the image of $$A$$ about the $$y$$-axis. Hence, if $$A\equiv (x,y)$$ then $$A'= (-x,y)$$.

Yes, we agree with this statement. Histogram is a form of representing the frequency distribution of any attribute or variables. It is a two dimensional figure that depict the frequency distribution of a continuous series by the means of rectangles. Here, adjacent rectangles are prepared consisting of length as well as breadth. The width and height of the rectangles determine the class interval and class frequency, respectively.
A frequency polygon, on the other hand, is a straight line curve obtained by joining the midpoints of the rectangular blocks of a histogram.

1. The diagrams should be simple.

2. Each diagram must be given a clear, concise and suitable title without damaging clarity.

3. A proper proportion between height and width must be maintained in order to avoid an unpleasant look.

4. Select a proper scale; it should be in even numbers or in multiples of five or ten.

e.g. 25,50, 75 or 10, 20, 30, 40, .... etc. But no fixed rule.

5. In order to clear certain points, always put footnotes.

6. An index, explaining different lines, shades and colors should be given.

7. Diagrams should be absolutely neat and clean.

A histogram refers to a visual representation of data using a set of vertical rectangles representing class frequencies. It is used to represent a continuous frequency distribution in a graphical manner by depicting classes in the form of rectangular bars.

A histogram is a visual representation of data using a set of vertical rectangles representing class frequencies. It is a graph and not a diagram. This is because it is a two-dimensional graph of frequency distribution that emphasises both the lengths and widths of the rectangles. Diagram, on the other hand, emphasises the length of the bars and not on the width. Histograms are also used to represent continuous frequency distributions. They are more technical and accurate in nature compared to diagrams.

The vertices of triangle$$ABC$$ are $$A(1,0), B(-5,0)$$ and $$C(-3,0)$$

Three given point are collinear.

Ordinate is $$y$$ co ordinate

Given $$PQ(l)+QR(b)=7$$ and $$PR+QS=10$$ and area of $$PQRS=lb$$

Midpoint $$(D)$$ of

Coordinate of point which is $$8$$ units right to $$P$$ is $$(7+8,0)=(15,0)$$.

Let the required point be $$(x, 0)$$. Then,

Cartesian coordinate system was developed by the famous French mathematician René Descartes. That is when, the co-ordinate geometry got shaped.

Let point $$(x_{1} , y_{1} ) $$ 
$$4y_{1} = x_{1}^{2} $$ ..........(i) 

$$d = \sqrt{(x_{1} - 0)^{2} + (y_{1} - 5)^{2}} $$ 

$$d^{2} = x_{1}^{2} + y_{1}^{2} + 25 - 10 y_{1} $$ 

$$d^{2} = 4y_{1} + y_{1}^{2} + 25 - 10y_{1} $$ 

Differentiate 
$$2d\ d' = 4 + 2y_{1} - 10 $$ 

For maximum & minimum 
$$4 + 2y_{1} - 10 = 0 $$ 
$$2y_{1} = 6 $$ 
$$y_{1} = 3 $$ 

$$4 \times 3 = x_{1}^{2} $$ 
$$x_{1} = \sqrt{12} $$ 

It is enough only one co-ordinate to explain a surface. So surface is one-dimensional.

1.The horizontal line is known as $$x-axis$$ and the vertical line is known as $$y-axis.$$

2. These parts are known as: $$1st, 2nd, 3rd$$ and $$4th$$ quadrants.

3. When these two lines intersect, it is known as intersecting point and also this is the origin point i.e $$(x,y) = (0,0)$$

Given $$\cfrac{AP}{BP}=\cfrac{2}{5}$$

$$x$$	$$3$$	$$3$$	$$3$$	$$3$$	$$3$$	$$3$$	$$3$$	$$3$$	$$3$$
$$y$$	$$-4$$	$$-3$$	$$-2$$	$$-1$$	$$0$$	$$1$$	$$2$$	$$3$$	$$4$$

Let the coordinates of point $$P$$ be $$(x, y).$$ 
The ratio in which $$P$$ divides $$A(-1,7)$$ and $$B(4,-3)$$ is $$2 : 3.$$ 
Using the section formula we have 
$$x=\dfrac{4 \times 2+3\times (-1)}{2+3}=\dfrac{8-3}{5}=1$$
$$y=\dfrac{2\times (-3)+3\times 7}{2+3}=\dfrac{-6+21}{5}=3$$
Thus, the coordinates of point $$P$$ are $$(1, 3).$$ 

Parametric form of straight line is.

Distance between A(1, 2, 3) and B(-1, -2, -1) is AB,

   $$= 6$$

Distance between B(-1, -2, -1) and C(2, 3, 2) is BC,

  

Distance between C(2, 3, 2) and D(4, 7, 6) is CD,

   $$= 6$$

The distance between A(1, 2, 3) and D(4, 7, 6) is AD,

  

Clearly, $$AB=CD$$ and $$BC=AD$$

Opposite sides are equal

Now, we will find the length of diagonals

The distance between A(1, 2, 3) and C(2, 3, 2) is AC,

  

Distance between B(-1, -2, -1) and D(4, 7, 6) is BD,

  

The distance between any two points $$(a, b, c)$$ and $$(m, n, o)$$ is given by,

Therefore,

Distance between O(0, 0, 0) and A(0, 1, 1) is

$$OA$$

  

Distance between O(0, 0, 0) and B(1, 0, 1) is OB,

  

Distance between O(0, 0, 0) and C(1, 1, 0) is

$$OC$$ 

  

Distance between A(0, 1, 1) and B(1, 0, 1) is 

$$AB$$ 

  

Distance between B(1, 0, 1) and C(1, 1, 0) is

$$BC$$

  

Distance between A(0, 1, 1) and C(1, 1, 0) is 

$$AC$$

  

Clearly,

$$AB = BC = AC = OA = OB = OC$$

All edges are equal,

Hence Proved

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

Distance between A(5, -1, 1) and B(7, -4, 7) is AB,

  = 7

Distance between B(7, -4, 7) and C(1, -6, 10) is BC,

   = 7

Distance between C(1, -6, 10) and D(-1, -3, 4) is CD,

   = 7

Distance between A(5, -1, 1) and D(-1, -3, 4) is AD,

   = 7

Clearly,

$$AB = BC = CD = AD$$

A three dimensional cartesian coordinate system with $$O$$ as the the origin is depicted in the figure.
We are given that the edges are parallel to neagtive $$X$$ and $$Y$$ axes, and the side of the cube is $$5$$ units.

Also, $$AB$$ is the line parallel to the $$Y-$$ axis, and hence the $$x$$ and $$z$$ coordinates of any point on the line $$AB$$ are constant.
The Coordinates of $$A:\ \ A(1,0,-1)$$
Therefore, the coordinates of $$B:\ \ B(1,5,-1)$$

$$AC$$ is the line parallel to $$Z$$ axis, and hence the $$x$$ and $$y$$ coordinates of any point on the line $$AC$$ are constant.
The Coordinates of $$A:\ \ A(1,0,-1)$$
Thus the coordinates of $$C:\ \ C(1,0,4)$$

$$AE$$ is the line parallel to $$X$$ axis, and hence the $$y$$ and $$z$$ coordinates of any point on the line $$AE$$ are constant.
The coordinates of $$A:\ \ A(1,0,-1)$$
Thus the coordinates of $$E:\ \ E(-4,0,-1)$$

$$GE$$ is the line parallel to $$Z$$ axis, and hence the $$x$$ and $$y$$ coordinates of any point on the line $$GE$$ are constant.
The coordinates of $$E:\ \ E(-4,0,-1)$$
Thus the coordinates of $$G:\ \ G(-4,0,4)$$

$$EF$$ is the line parallel to $$Y$$ axis, and hence the $$x$$ and $$z$$ coordinates of any point on the line $$EF$$ are constant.
The coordinates of $$E:\ \ E(-4,0,-1)$$
Thus the coordinates of $$F:\ \ F(-4,-5,-1)$$

$$GH$$ is the line parallel to $$Y$$ axis, and hence the $$x$$ and $$z$$ coordinates of any point on the line $$GH$$ are constant.
The coordinates of $$G:\ \ G(-4,0,4)$$
Thus the coordinates of $$H:\ \ H(-4,-5,4)$$

Hindi and English is the language in which the largest no. of newspapers were published from the graph.

From the given diagram, we observed that the coordinates of following points$$:-$$
$$O(0,0)$$
$$ X(0,2)$$
$$Y(2,2)$$
$$Z(2,0)$$
$$A(4,5)$$
$$B(7,5)$$
$$C(6,3)$$
$$D(3,3)$$
$$P(7,4)$$
$$Q(9,5)$$
$$R(9,3)$$

Monthly School Fee	30-60	60-90	90-120	120-150	150-180	180-210	210-240
Number of schools	5	12	14	18	10	9	4

The answer is False as the ration would be 50:400 = 1:8

After observing the graph, we conclude that,

The point where the two axes intersect is called the Origin or $$(0, 0)$$ point.

On the scale of $$1$$ unit length $$= 10$$ crore, the bar of length $$6$$ units will represent $$60$$ crore and of $$7.5$$ units will represent $$75$$ crore 

The $$x$$-coordinate of any point lying on the $$y$$-axis will be zero .

All points with y-coordinate as zero lie on the x-axis

The graph that we get follows the line y = 3x - 1 for x between 1 to 5.

The three vertices of a parallelogram ABCD are given as $$A(3, -1, 2), B(1, 2, -4)$$ and $$C(-1, 1, 2).$$ Let the coordinates of the fourth vertex be $$D(x, y, z).$$
We know that the diagonals of a parallelogram bisect each other.
Therefore, in parallelogram ABCD, AC and BD bisect each other.
$$\therefore$$ Mid - point of AC = Mid-point of BD
$$\Rightarrow \left ( \dfrac{3 - 1}{2}, \dfrac{-1 + 1}{2}, \dfrac {2 + 2}{2} \right ) = \left ( \dfrac{x + 1}{2}, \dfrac{y + 2}{2}, \dfrac{z - 4}{2} \right )$$
$$\Rightarrow (10, 2) = \left ( \dfrac{x + 1}{2}, \dfrac{y + 2}{2}, \dfrac{z - 4}{2} \right )$$
$$\Rightarrow \dfrac{x + 1}{2} = 1, \dfrac{y + 2}{2} = 0,$$ and $$\dfrac{z - 4}{2} = 2$$
$$\Rightarrow x = 1, y = 2$$ and $$z = 8$$
Thus, the coordinates of the fourth vertex are $$(1, -2, 8).$$

Item	Name of items sold store $$A$$	Name of items sold store $$B$$
Grill	$$40$$	$$20$$
Toaster	$$35$$	$$15$$
Oven	$$30$$	$$30$$
Blender	$$40$$	$$30$$
Coffee maker	$$35$$	$$40$$

The double bar graph of the given data is shown above.

The graph represents number of boys and girls, going to school using different modes of transport.

Items	Expenditure in (Rs.)	Central Angle = (Item/Total) x 360
Cement	1800	(1800/10000) x 360 = 64.8
Bricks	2000	(2000/10000) x 360 = 72
Iron	4200	(4200/10000) x 360 = 151.2
Labour	2000	(2000/10000) x 360 = 72
Total	10000	360

Co-ordinate of points $$ \mathrm{A}, \mathrm{B}, \mathrm{C} $$ are $$(6,-1),(1,3)$$ and $$ (\mathrm{x}, 8) $$ respectively.
According to question

The mode of transport, boys using the most is bicycle 

The mode of transport, girls using more than the boys is school bus

First of all we draw co-ordinate axis XOX' and YOY' and mark the points $$ \mathrm{A}(-2,0), \mathrm{B}(2,0), \mathrm{C}(2,2) \mathrm{D}(0,4) $$ and $$ \mathrm{E}(-2,2) $$ then we get pentagon.

Transformation that map shape A onto shape B is rotation.

Items	Expenditure (%)	Central Angle = (Expenditure/Total x 100)
Labour	20	72⁰
Bricks	15	54⁰
Cement	25	90⁰
Steel	10	36⁰
Timber	20	72⁰
Supervision	10	36⁰
Total	100	360⁰

Given data shows the sales of a commodity during the years 2000 to 2006

year	2000	2001	2002	2003	2004	2005	2006
Sale  (in lakhs of Rs.)	1.5	1.8	2.4	3.2	5.4	7.8	8.6

Transformation that map shape A onto shape C is rotation.

The transformation that map shape A onto shape D is translation.