This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the input value.

Since the function is in the form $\displaystyle f(x)$ and the question asks to calculate $\displaystyle f(6)$ we know that the input value is six.

$\displaystyle f(6)\rightarrow x=6$

Step 2: Given the function, input the desired $\displaystyle x$ value. 

In step one the input value of six was found. Now substitute 6 in for every$\displaystyle x$ in the function.

$\displaystyle f(x)=x^2-2x$

$\displaystyle f(6)=6^2-2(6)$

$\displaystyle f(6)=36-12$

$\displaystyle f(6)=24$

Step 3: Verify solution by graphing the function.

To find the function value on a graph go to $\displaystyle x=6$ and find the y value that lies on the graph at $\displaystyle x=6$. Looking at the graph above when $\displaystyle x=6$, $\displaystyle y=24$.

Recall that $\displaystyle f(x)=y \rightarrow f(6)=24$, thus verifying the solution found in step 2.

To solve this problem, first recall the equation of a line.

$\displaystyle y=mx+b$

where

$\displaystyle \\m=\text{slope} \\y=\text{y-intercept}$

Remember that slope is the rate of change that occurs in a function and that the $\displaystyle y$-intercept is the $\displaystyle y$ value corresponding to an $\displaystyle x$ value equalling zero.

Now, identify what is known.

"Billy buys a tomato plant that is 4 inches tall. With regular watering the plant grows 3 inches a year."

Since the height of Billy's plant is 4 inches tall when he gets it, at time being zero. It is also stated that the plant grows 3 inches per year, this is the rate of change of the plant's height. Writing the known information in mathematical terms results as follows.

$\displaystyle h(t)=3t+4$

In this case our 

$\displaystyle \\y=h(t)=\text{plant's height as a function of time} \\t=years \\m=3 \\b=4$

Therefore, the $\displaystyle y$-intercept represents the starting height of the plant which is 4 inches.

To solve this problem, first recall the equation of a line.

$\displaystyle y=mx+b$

where

$\displaystyle \\m=\text{slope} \\y=\text{y-intercept}$

Remember that slope is the rate of change that occurs in a function and that the $\displaystyle y$-intercept is the $\displaystyle y$ value corresponding to an $\displaystyle x$ value equalling zero.

Now, identify what is known.

"Billy buys a tomato plant that is 4 inches tall. With regular watering the plant grows 3 inches a year."

Since the height of Billy's plant is 4 inches tall when he gets it, at time being zero. It is also stated that the plant grows 3 inches per year, this is the rate of change of the plant's height. Writing the known information in mathematical terms results as follows.

$\displaystyle h(t)=3t+4$

In this case our 

$\displaystyle \\y=h(t)=\text{plant's height as a function of time} \\t=years \\m=3 \\b=4$

Therefore, the slope represents the rate of change in the height of the plant which is 3 inches per year.

First calculate the slope of the line using the slope formula.

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$

Substituting in the known information.

$\displaystyle \\(x_1, y_1)=(1,2) \\(x_2,y_2)=(4,5)$

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-2}{4-1}=\frac{3}{3}=1$

Now use point slope form to find the equation of the line passing through these points.

$\displaystyle \\y-y_1=m(x-x_1) \\y-2=1(x-1) \\y-2=x-1 \\y=x+1$

Now identify which point lies on the line.

$\displaystyle \\(9,10)\Rightarrow10=9+1 \\(3,2)\Rightarrow 2\neq3+1 \\(0,0)\Rightarrow0\neq0+1 \\(2,2)\Rightarrow 2\neq2+1 \\(5,3)\Rightarrow 3\neq5+1$

Therefore, the point that lies on the line is (9,10)

To solve this problem first manipulate the given equation to solve for $\displaystyle c$.

$\displaystyle \frac{c}{d}=6$

Multiply by $\displaystyle d$ on both sides. The $\displaystyle d$'s on the left hand side of the equation will cancel out.

$\displaystyle d\cdot\frac{c}{d}=6\cdot d$

$\displaystyle c=6d$

Now to calculate $\displaystyle \frac{3d}{c}$, substitute the value for $\displaystyle c$ that was just found, into the denominator and simplify by canceling out common factors.

$\displaystyle \frac{3d}{c}=\frac{3d}{6d}=\frac{3}{6}=\frac{1\cdot 3}{2\cdot 3}=\frac{1}{2}$

$\displaystyle y=x^2-5x+6$

First we group terms

$\displaystyle y=x^2-5x+6=(x^2-5x)+6$

Now we want to have a perfect square, so we add $\displaystyle \left(\frac{b}{2}\right)^2$, and we subtract $\displaystyle \left(\frac{b}{2}\right)^2$, so now it looks like

$\displaystyle \left(x^2-5x+\left(\frac{-5}{2}\right)^2\right)-\left(\frac{-5}{2}\right)^2+6$

$\displaystyle (x^2-5x+6.25)-6.25+6$

Simplify to get

$\displaystyle y=(x-2.5)^2-0.25$

$\displaystyle f(x) = x^2 + 32x -12$

$\displaystyle f(4) = 4^2 + 32(4) -12$

$\displaystyle = 16 + 128 -12$

$\displaystyle = 132$

We can use point slope form to determine the equation of a line that fits the data. 

Point slope form is $\displaystyle y-y_0=m(x-x_0)$, where $\displaystyle (x_0, y_0)$, and $\displaystyle m$ is the slope, where $\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$.

Let $\displaystyle y_2=6$, $\displaystyle y_1=3$, $\displaystyle x_2=3$, and $\displaystyle x_1=2$.

$\displaystyle m=\frac{6-3}{3-2}=3$

If we do this for every other point, we will see that they have the same slope of $\displaystyle 3$.

$\displaystyle y-y_0=3(x-x_0)$

Now let $\displaystyle x_0=0$, and $\displaystyle y_0=-3$.

$\displaystyle y-(-3)=3(x-0)$

$\displaystyle y+3=3x$

$\displaystyle y=3x-3$

If we use the equation that solved for previously, $\displaystyle y=5x+2$, we just plug in $\displaystyle 15$ for $\displaystyle x$.

$\displaystyle y=5(15)+2=75+2=77$

The answer is, "The estimated increase in the average number of college graduates each year" because the equation is a model, which is an estimation.