Introduction to Graphs - The Cartesian Plane and Coordinates

Identify the location (Quadrant or Axis) of the following points: $A(4, 5)$, $B(-2, 3)$, $C(-1, -5)$, $D(3, -2)$, and $E(0, 7)$.

1. Point $A(4, 5)$: Both $x > 0$ and $y > 0$, so it is in Quadrant I.
2. Point $B(-2, 3)$: Here $x < 0$ and $y > 0$, so it is in Quadrant II.
3. Point $C(-1, -5)$: Both $x < 0$ and $y < 0$, so it is in Quadrant III.
4. Point $D(3, -2)$: Here $x > 0$ and $y < 0$, so it is in Quadrant IV.
5. Point $E(0, 7)$: Since the x-coordinate is $0$, the point lies on the y-axis.

To determine the location, check the signs of the $x$ and $y$ coordinates. If a coordinate is $0$, the point lies on an axis rather than in a quadrant.

A point $P$ is $4$ units to the left of the y-axis and $6$ units above the x-axis. Find its coordinates.

1. '4 units to the left of the y-axis' means the x-coordinate (abscissa) is negative: $x = -4$.
2. '6 units above the x-axis' means the y-coordinate (ordinate) is positive: $y = 6$.
3. Combining these into an ordered pair $(x, y)$, we get $P(-4, 6)$.

Direction matters: 'Left' and 'Down' indicate negative values; 'Right' and 'Up' indicate positive values relative to the origin.

1. Starting at the origin, move $2$ units right and $3$ units up to plot $A(2, 3)$.
2. Move $3$ units left and $3$ units up to plot $B(-3, 3)$.
3. Locate $-2$ on the $y$-axis to plot $C(0, -2)$.
4. Joining $A$ to $B$, $B$ to $C$, and $C$ to $A$ forms a triangle.

By connecting three non-collinear points in a plane, a triangle is always formed. Here, the vertices are in different quadrants and on an axis.

1. Plot $P(-2, -2)$ in the third quadrant and $Q(2, 2)$ in the first quadrant.
2. Draw a straight line through these points.
3. Observation: The line passes exactly through $(0, 0)$.
4. For every point on this line, the $x$-coordinate is equal to the $y$-coordinate ($x = y$).

The line represents the linear equation $y = x$. Since the coordinates are equal at every plotted point, it must pass through the origin where $0 = 0$.