To answer this question, we need to correctly identify where to plug in our given values and solve for $\displaystyle a$.

Points on a graph are written in coordinate pairs. These pairs show the $\displaystyle x$ value first and the $\displaystyle y$ value second. So, for this data:

$\displaystyle (1,2a)$ means that $\displaystyle 1$ is the $\displaystyle x$ value and $\displaystyle 2a$ is the $\displaystyle y$ value.

We must now plug in our $\displaystyle x$ and $\displaystyle y$ values into the original equation and solve. Therefore:

$\displaystyle y=-4x^{2}+6\rightarrow 2a=-4(1)^{2}+6$

We can now begin to solve for $\displaystyle a$ by adding up the right side and dividing the entire equation by $\displaystyle 2$.

$\displaystyle 2a=-4(1)^{2}+6\rightarrow 2a=-4+6\rightarrow2a=2$

$\displaystyle 2a=2\rightarrow \frac{2a}{2}=\frac{2}{2}\rightarrow a=1$

Therefore, the value of $\displaystyle a$ is $\displaystyle 1$.

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 $\displaystyle (4,-1)$.

Because points on a graph are written in the form of $\displaystyle \small (x,y)$, and the point given was $\displaystyle \small (3, 5b)$, this means that $\displaystyle \small x=3$ and $\displaystyle \small y=5b$.

In order to solve for $\displaystyle \small b$, these values for $\displaystyle \small x$ and $\displaystyle \small y$ must be plugged into the given equation. This gives us the following:

$\displaystyle \small y=2x^3-9$

$\displaystyle \small 5b=2(3)^3-9$

We then solve the equation by finding the value of the right side, then dividing the entire equation by 5, as follows:

$\displaystyle \small 5b=2(27)-9$

$\displaystyle 5b=54-9$

$\displaystyle 5b=45$

$\displaystyle \frac{5b}{5}=\frac{45}{5}$

$\displaystyle b=9$

Therefore, the value of $\displaystyle \small b$ is $\displaystyle \small 9$.