A kosarad üres
The Cartesian coordinate system, point sets
With the help of the Cartesian coordinate system an ordered pair of real numbers is assigned to each point of the plane. The first term of the number pair, in other words its abscissa, gives the signed distance of the point measured from the \latex{y}-axis, the second term, in other words its ordinate, gives the signed distance of the point measured from the \latex{x}-axis. The converse is also true, a single point of the plane belongs to every ordered number pair.
Example 1
Let us plot the following points in the Cartesian coordinate system:
\latex{A(1; 2)}, \latex{B(-2; 1)}, \latex{C(-3; -2)}, \latex{D(2; -2)}, \latex{E(0; -3)}, \latex{F(2; 0)}.
Figure 1
\latex{ 2 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3}
\latex{ -1 }
\latex{ -2 }
\latex{ B }
\latex{ A }
\latex{ F }
\latex{ D }
\latex{ E }
\latex{ C}
\latex{ y }
\latex{ x }
Solution
The representation of the points is shown in Figure 1.
Example 2
Let us read the coordinates of points \latex{ P, Q, R, S } fom Figure 2.
Figure 2
\latex{ R }
\latex{ S }
\latex{ Q }
\latex{ P }
\latex{ y }
\latex{ x }
Solution
The coordinates of the points are:
\latex{P(-2; -1)}, \latex{Q(2; -4)}, \latex{R(-4; 3)}, \latex{S(3; 4)}.
⯁ ⯁ ⯁
The ordinate of the points of the \latex{x}-axis is 0, i.e. \latex{y = 0}. According to this the points of the \latex{y}-axis can be described as: \latex{x = 0}.
Example 3
Where are those points in the plane for the coordinates of which the following are fulfilled: \latex{x\gt0} and \latex{y\gt0?}
Solution
In Figure 3 the sector of the plane shaded darker fulfil the two conditions. This range is called the first quadrant.
first
quadrant
quadrant
second
quadrant
quadrant
third
quadrant
quadrant
fourth
quadrant
quadrant
Figure 3
\latex{ 1 }
\latex{ 1 }
\latex{ y }
\latex{ x }
⯁ ⯁ ⯁
According to the conventions the points of the second quadrant are described by inequalities \latex{x\lt 0}, \latex{y\gt 0}, the third quadrant is described by \latex{x\lt 0}, \latex{y\lt 0} and finally the points of the fourth quadrant are described by inequalities \latex{x\gt 0}, \latex{y\lt 0}.
The equality can be allowed in every inequality; it means that the corresponding borderline is also part of the quadrant.
For example the point set described by inequalities \latex{x \leq0}, \latex{y \gt 0} consists of the points of the second quadrant with the positive part of the \latex{ y }-axis inclusive.
Example 4
Where are those points in the plane which are equidistant from the x-axis and the
y-axis?
y-axis?
\latex{y=x}
\latex{y=-x}
Figure 4
\latex{ 1 }
\latex{ 1 }
\latex{x}
\latex{y}
Solution
Let us examine some points in the Cartesian coordinate system. The points equidistant from two intersecting straight lines in the plane are the angle bisectors of the angle defined by the two straight lines. So the points which are equidistant from the x-axis and the y-axis are the points of the two angle bisectors.
These points can be described for example as: \latex{y=x} and \latex{y=-x}, or summarised as \latex{\left|y\right|=\left|x\right|}. (Figure 4)
Example 5
For the coordinates of which points is the following inequality true: \latex{y\leq x?}
Solution
Firstly let us check a few points again. We know by now that for the angle bisector going through the first and the third quadrant it is true that \latex{y = x}. If starting from any of its points we go along the y-axis into the negative direction (downwards), then the x-coordinate does not change, while the y-coordinate decreases. (Figure 5)
So inequality \latex{y\leq x} is fulfilled for the coordinates of the points of the angle bisector and for the coordinates of the points of the plane below the angle bisector (the line \latex{y = x}).
\latex{y=x}
Figure 5
\latex{ y }
\latex{ x }
\latex{ 1 }
\latex{ 1 }
Example 6
Where are those points in the plane for the coordinates of which inequalities \latex{\left|x\right|\leq2} and \latex{\left|y\right|\leq2} are fulfilled?
\latex{\left|x\right|\leq2 \Leftrightarrow} \latex{-2\leq x\leq2}
Figure 6
\latex{ -2 }
\latex{ +2 }
\latex{ x }
Solution
Inequality \latex{\left|x\right|\leq2} can be rewritten as \latex{-2\leq x \leq2}. The points with coordinates satisfying the first condition are between two straight lines parallel with the y-axis, the points with coordinates satisfying the second condition are between two straight lines parallel with the x-axis. The points the coordinates of which satisfy both inequalities are the common points of the two bars, i.e. the internal points of a square and the points of the sides of the square. (Figure 7)
Figure 7
\latex{ 2 }
\latex{ 2 }
\latex{ -2 }
\latex{ -2 }
\latex{ y }
\latex{ x }
Example 7
Give the set of points \latex{P(x; y)} in the plane for the coordinates of which the following inequality is fulfilled: \latex{\left|y-x\right|\leq 1}.
\latex{y=x+1}
\latex{y=x-1}
Figure 8
\latex{ -1 }
\latex{ -1 }
\latex{ 1 }
\latex{ 1 }
\latex{ y }
\latex{ x }
Solution
Based on the definition of the absolute value the inequality can also be written as follows: \latex{-1\leq y-x\leq 1}, or by adding \latex{ x } to all three expressions: \latex{x-1\leq y\leq x+1}. The equation \latex{y=x+1} is satisfied by the coordinates of the points of the straight line intersecting the \latex{y}-axis at point \latex{(0; 1)} and parallel with the angle bisector described by equation \latex{y = x}. Similarly the equality \latex{y = x-1} is fulfilled for the coordinates of the points of the straight line intersecting the \latex{y}-axis at point \latex{(0; -1)} and parallel with the above. The inequality in question is fulfilled for the coordinates of the points of the track between the two parallel straight lines shown in the figure and for the coordinates of the points of the straight lines bounding the track. (Figure 8)
Exercises
{{exercise_number}}. Plot the following points: \latex{A(-3; -2)}, \latex{B(0; -5)}, \latex{C(3; -1)}, \latex{D(2; -2)}, \latex{E(4; 4)}, \latex{F(-4; -4)}.
{{exercise_number}}. Plot the points in the plane the coordinates of which satisfy the following equations.
- \latex{x=3}
- \latex{y=-2}
- \latex{y=-x}
- \latex{y=x+2}
{{exercise_number}}. Plot the points in the plane the coordinates of which fulfil the following inequalities.
- \latex{x\leq3}
- \latex{y\geq-2}
- \latex{-2\leq x\leq 3}
- \latex{1\leq \left|y\right| \leq 2}
{{exercise_number}}. Plot the points \latex{P(x; y)} in the plane the coordinates of which fulfil the following conditions.
- \latex{x\times y=0}
- \latex{\left|x\right|\leq\left|y\right|}
- \latex{\left|x\right|+\left|y\right|\leq 1}
- \latex{\left|x-y\right|+\left|x+y\right|\leq2}
{{exercise_number}}. The following two point sets are given:
\latex{A=}{\latex{{ (x; y)| x\in \R }} and \latex{\;y\in \R \;and\; x\geq 1}} and
\latex{B=}{\latex{{ (x; y)| x\in \R }} and \latex{\;y\in \R \;and\; y\leq -1}}.
\latex{B=}{\latex{{ (x; y)| x\in \R }} and \latex{\;y\in \R \;and\; y\leq -1}}.
Plot the following sets in the coordinate system.
- \latex{A \cap B}
- \latex{A \cup B}
- \latex{A\backslash B}
- \latex{B\backslash A}
{{exercise_number}}. Plot the points \latex{P(x; y)} in the plane the coordinates of which fulfil the following conditions simultaneously.
- \latex{\begin{aligned}x-y&\gt5\\ 3x+4y&\gt4\end{aligned}}
- \latex{\begin{aligned}x-1&\gt y\\2x-3y&\gt6\end{aligned}}
- \latex{\begin{aligned}x+y&\gt1\\2x-y&\lt1\\x &\gt-1\end{aligned}}
- \latex{\begin{aligned}\left|x\right|+\left|y\right|&\lt2\\\left|x\right|-\left|y\right|&\gt 1\end{aligned}}
Quiz
A flea is jumping around in the coordinate system. It starts from the origin and jumps \latex{ 1 } unit to the right or up in every \latex{second}.
- In how many \latex{seconds} can it reach point \latex{(3; 5)} when counted from the start?
- In how many different ways can it reach point \latex{(3; 5)?}
