Mrs. Smith's 8th grade class had a quiz last week. The graph below depicts the number of questions students got incorrect on their quiz and their corresponding quiz grade. In other words, the graph in this particular question is a dot plot and the question asks to find a correlation if one exists.

Recall that a correlation is a trend seen in the data. Graphically, trends can be either:

I. Positive 

II. Negative

III. Constant

IV. No trend

For a trend to be positive the x and y variable both increase. A trend is negative when the y variable (dependent variable) decreases as the x variable (independent variable) increases. A constant trend occurs when the y variable stays the same as the x variable increases. No trend exists when the data appears to be scattered with no association between the x and y variables.

Examining the graph given it is seen that the x variable is the number of questions missed and the y variable is the overall score on the quiz. It is seen that as the number of questions missed increases, the overall score on the quiz decreases. This describes  a negative trend.

In other words, the graph depicts a negative correlation.

Given the graph of record sales, to find the fraction of records that were sold in 2004 to 2010 first identify the record sales in 2004 and the record sales in 2010.

Examining the graph,

Record sales in 2004:14 million

Record sales in 2010: 13 million

From here, to find the fraction of records sold during this time period, use the following formula.

Using the distance formula, calculate the distance from each of these points to the origin, (0, 0). While each answer choice has coordinates that add up to seven, (-1, 8) is the coordinate pair that produces the largest distance, namely , or approximately 8.06.

On the coordinate plane, a point with a negative -coordinate and a positive -coordinate lies in the upper left quadrant—Quadrant II.

The slope of the line is  and the y-intercept is .

The equation of the line is . 

This is the parent graph of . Since the graph in question is negative, then we flip the quadrants in which it will approach infinity. So the graph of  will start in quadrant 2 and end in 4. 

It is necessary to find the coordinates of the vertices of the triangle, each of which is the intersection of two of the three lines.

The intersection of the lines of the equations  and  can be found by noting that, by substituting   for  in the latter equation, , making the point of intersection .

The intersection of the lines of the equations  and  can be found by substituting   for  in the latter equation and solving for :

This point of intersection is .

The intersection of the lines of the equations  and  can be found by substituting   for  in the latter equation and solving for :

Since , , and this point of intersection is .

The lines in question are graphed below, and the triangle they bound is shaded:

We can take the horizontal side as the base of the triangle; its length is the difference of the -coordinates:

The height is the vertical distance from this side to the opposite side, which is the difference of the -coordinates:

The area is half their product:

The value of the slope (m) is rise over run, and can be calculated with the formula below:

The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall. 

The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end. 

From this information we know that we can assign the following coordinates for the equation:

 and 

Below is the solution we would get from plugging this information into the equation for slope:

This reduces to 

Looking at the graph, it is seen that the line passes through the points (-8,-5) and (8,5).

The slope of a line through the points  and  can be found by setting 

:

in the slope formula:

This question tests one's ability to recognize algebraic characteristics of a graph. This particular question examines a linear function.