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 \(\displaystyle a+bi\) can be represented graphically by the point \(\displaystyle (a,b)\).

Here, we are given the graph and asked to write the corresponding expression.

\(\displaystyle 3-2i\) not only correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, it also includes the necessary \(\displaystyle i\). 

\(\displaystyle 3-2\) correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, but fails to include the necessary \(\displaystyle i\).

\(\displaystyle -2+3i\) misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.

\(\displaystyle -2+3\) misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.  It also fails to include the necessary \(\displaystyle i\).

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 \(\displaystyle a+bi\) can be represented graphically by the point \(\displaystyle (a,b)\).

Here, we are given the complex number \(\displaystyle 4-i\) and asked to graph it.  We will represent the real part, \(\displaystyle 4\), on the x-axis, and the imaginary part, \(\displaystyle -i\), on the y-axis.  Note that the coefficient of \(\displaystyle i\) is \(\displaystyle -1\); this is what we will graph on the y-axis.  The correct coordinates are \(\displaystyle (4,-1)\).

The \(\displaystyle x\)-coordinate(s) of the \(\displaystyle x\)-intercept(s) are the real solution(s) to the equation \(\displaystyle f(x) = x^{2}+4x+7 = 0\). We can use the quadratic formula to find any solutions, setting \(\displaystyle a=1, b=4, c=7\) - the coefficients of the expression.

An examination of the discriminant \(\displaystyle b^{2} - 4ac\), however, proves this unnecessary.

\(\displaystyle b^{2} - 4ac = 4^{2} - 4\cdot 1\cdot 7 = -12\)

The discriminant being negative, there are no real solutions, so the parabola has no  \(\displaystyle x\)-intercepts.

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  \(\displaystyle -5\)  and the imaginary component is  \(\displaystyle 3\),  so this is the equivalent of plotting the point  \(\displaystyle (-5,3)\)  on a set of Cartesian axes.  Plotting the complex number on a set of real-imaginary axes, we move  \(\displaystyle 5\)  to the left in the x-direction and  \(\displaystyle 3\)  up in the y-direction, which puts us in the second quadrant, or in terms of Roman numerals:

\(\displaystyle Answer: II\)

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:

\(\displaystyle 9-4i\)

We are essentially doing the same as plotting the point  \(\displaystyle (9,-4)\)  on a set of Cartesian axes.  We move  \(\displaystyle 9\)  units right in the x direction, and  \(\displaystyle 4\)  units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:

\(\displaystyle Answer:IV\)

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:

\(\displaystyle -13-5i\)

We are essentially doing the same as plotting the point  \(\displaystyle (-13,-5)\)  on a set of Cartesian axes.  We move  \(\displaystyle 13\)  units left of the origin in the x direction, and  \(\displaystyle 5\)  units down from the origin in the y direction, which puts us in the third quadrant, or in terms of Roman numerals:

\(\displaystyle Answer:III\)

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:

\(\displaystyle 23+14i\)

We are essentially doing the same as plotting the point  \(\displaystyle (23,14)\)  on a set of Cartesian axes.  We move  \(\displaystyle 23\)  units right of the origin in the x direction, and  \(\displaystyle 14\)  units up from the origin in the y direction, which puts us in the first quadrant, or in terms of Roman numerals:

\(\displaystyle Answer:I\)

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.

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. 

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. 

\(\displaystyle a^{2}+b^{2}=c^{2}\ so\ 3^{2}+4^{2}=c^{2}\)

\(\displaystyle 9+16=c^{2}\)

\(\displaystyle 25=c^{2}\)

\(\displaystyle c\(the\ absolute\ value\ or\ distance\from\ the\ origin)=5\)