Coordinate Geometry - Geometry

Complex numbers can be represented on the coordinate plane by mapping the real part to the x-axis and the imaginary part to the y-axis. For example, the expression can be represented graphically by the point .

Here, we are given the complex number and asked to graph it. We will represent the real part, , on the x-axis, and the imaginary part, , on the y-axis. Note that the coefficient of is ; this is what we will graph on the y-axis. The correct coordinates are .

When plotting a complex number, we use a set of real-imaginary axes in which the x-axis is represented by the real component of the complex number, and the y-axis is represented by the imaginary component of the complex number. The real component is and the imaginary component is , so this is the equivalent of plotting the point on a set of Cartesian axes. Plotting the complex number on a set of real-imaginary axes, we move to the left in the x-direction and up in the y-direction, which puts us in the second quadrant, or in terms of Roman numerals:

The -coordinate(s) of the -intercept(s) are the real solution(s) to the equation . We can use the quadratic formula to find any solutions, setting - the coefficients of the expression.

An examination of the discriminant , however, proves this unnecessary.

The discriminant being negative, there are no real solutions, so the parabola has no -intercepts.

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units right in the x direction, and units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units left of the origin in the x direction, and units down from the origin in the y direction, which puts us in the third quadrant, or in terms of Roman numerals:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units right of the origin in the x direction, and units up from the origin in the y direction, which puts us in the first quadrant, or in terms of Roman numerals:

In the complex plane, the x-axis represents the real component of the complex number, and the y-axis represents the imaginary part. The point shown is (8,3) so the real part is 8 and the imaginary part is 3, or 8+3i.

The definition of Absolute Value on a coordinate plane is the distance from the origin to the point.

When graphing this complex number, you would go 3 spaces right (real axis is the x-axis) and 4 spaces down (the imaginary axis is the y-axis).

This forms a right triangle with legs of 3 and 4.

To solve, plug in each directional value into the Pythagorean Theorem.

For the answer we must know that the x-axis is the real axis and the y-axis is the imaginary axis. We can see that we have gone 2 spaces in the x-axis or real direction and -3 spaces in the y-axis or imaginary direction to give us the answer 2-3i.

In the complex plane, the imaginary axis matches the y-axis and the real axis matches the x-axis.

When graphing complex numbers, we go left or right to graph the real number and then up or down to graph the complex number.

To subtract two complex numbers, subtract the real parts and subtract the imaginary parts:

On the complex plane, a complex number whose real part is positive is right of the imaginary (vertical) axis; a complex number whose imaginary coefficient is negative is below the real (horizontal axis). is three units right of and five units below the origin; this is point D.

To add two complex numbers, add the real parts and add the imaginary parts:

On the complex plane, a complex number whose real part is negative is left of the imaginary (vertical) axis; a complex number whose imaginary coefficient is positive is above the real (horizontal axis). is three units left of and five units above the origin; this is point A.

On the complex plane, the real and imaginary parts of a point are equal to the horizontal and vertical distances, respectively, from the origin to the point, with the right and upward directions representing positive quantities and the left and downward directions representing negative quantities. Refer to the diagram below, which shows the horizontal and vertical distances from the origin of both points.

is six units right of the origin, representing real part 6, and eight units above the origin, representing imaginary part . is two units right of the origin, representing real part 2, and five units below the origin, representing imaginary part . Therefore,

We are asked to find .

To add two complex numbers, add the real parts and add the imaginary parts:

On the complex plane, the real and imaginary parts of a point are equal to the horizontal and vertical distances, respectively, from the origin to the point, with the right and upward directions representing positive quantities and the left and downward directions representing negative quantities. Refer to the diagram below, which shows the horizontal and vertical distances from the origin of both points.

is six units right of the origin, representing real part 6, and eight units above the origin, representing imaginary part . is two units right of the origin, representing real part 2, and five units below the origin, representing imaginary part . Therefore,

We are asked to find .

To subtract two complex numbers, subtract the real parts and subtract the imaginary parts:

On the complex plane, the real and imaginary parts of a point are equal to the horizontal and vertical distances, respectively, from the origin to the point, with the right and upward directions representing positive quantities and the left and downward directions representing negative quantities. Refer to the diagram below, which shows the horizontal and vertical distances from the origin of both points.

is six units right of the origin, representing real part 6, and eight units above the origin, representing imaginary part . is two units right of the origin, representing real part 2, and five units below the origin, representing imaginary part . Therefore,

To multiply two complex numbers, use the FOIL method as if you were multiplying two binomials.

F(irst):

O(uter):

I(nner):

L(ast):

By definition, , so the "L" quantity simplifies:

.

Add these quantities, collecting real parts (F and L) and imaginary parts (O and I):

If the graph of is concave upward, then .

If the graph does not intersect the -axis, then has no real solution, and the discriminant is negative:

For the parabola to have both characteristics, it must be true that and , but these two events are mutually exclusive. Therefore, the parabola cannot exist.

The line of symmetry of a horizontal parabola is a horizontal line, the equation of which takes the form for some . The line of symmetry passes through the vertex, which here is , so the equation must be .

The line of symmetry of a vertical parabola is a vertical line, the equation of which takes the form for some . The line of symmetry passes through the vertex, which here is , so the equation must be .

A parabola with its -intercepts at and at has as its equation

for some nonzero . If this is multiplied out, the equation can be rewritten as

or, simplified,

The sign of quadratic coefficient determines whether it is concave upward or concave downward. We do not have the sign or any way of determining it.

The -coordinate of the -intercept is the contant, , but without knowing , we have no way of knowing .

The -coordinate of the vertex of is the value . since , this expression becomes

The -coordinate is

,

but without knowing , this coordinate, and the vertex itself, cannot be determined.

The line of symmetry is the line ; this value was computed to be equal to 6, so the line can be determined to be .