EXTENDED RESPONSE The table shows how values of a painting increased over ten years. Part A: Select and create a display that shows the relationship between years and the value of the painting. Justify your reasoning. Part B: Write a conclusion based on your graph. Part C: Use the graph to predict what the value of the painting will be in 2018. Year: 2005, 2006, 2007, 2008, and 2009. Value: $350, $650, $950, $1,200, and $1,551. Year: 2010, 2011, 2012, 2013, and 2014. Value: $1,851, $2,151, $2,451, $2,752, and $3,052.
Part A: A line graph would be the best display to show the relationship between years and the value of the painting. The x-axis would represent the years, and the y-axis would represent the value of the painting. Each year's value would be plotted and connected with a line.
Part B: Based on the graph, it can be seen that the value of the painting steadily increases over time. The value seems to increase at a faster rate in the later years (2010-2014) as opposed to the earlier years (2005-2009).
Part C: Using the graph, it can be predicted that the value of the painting in 2018 would be approximately $3,352. This prediction is based on the steady increase in value shown in the graph and assuming that the trend continues.
Part A:
To display the relationship between years and the value of the painting, a line graph would be the most appropriate choice. The x-axis can represent the years (starting from 2005 and ending in 2014), while the y-axis can represent the value of the painting (ranging from $350 to $3,052). Each data point from the table can be plotted as a coordinate on the graph, with the x-coordinate representing the year and the y-coordinate representing the value.
Justification: A line graph is suitable for this situation because it allows for the visualization of the trend in the data over time. By connecting the data points with a line, it becomes easier to observe any patterns or changes in the value of the painting as the years progress.
Part B:
Based on the line graph, we can draw the following conclusions:
1. The value of the painting has steadily increased over the ten-year period.
2. The rate of increase appears to be accelerating. From 2005 to 2009, the value increased by $1,201, while from 2010 to 2014, the value increased by $1,201 as well.
3. There is a positive correlation between the years and the value of the painting.
Part C:
Using the line graph, we can predict the value of the painting in 2018. We can extrapolate the pattern based on the trend observed in the graph. Since we know that each five-year increment resulted in an increase of $1,201, we can assume that the same rate of increase will continue.
From 2014 to 2018, there are four years, which is 80% of a five-year increment ($1,201). Therefore, we can estimate that the value of the painting in 2018 will be approximately 80% of $1,201 above the value in 2014.
$1,201 * 0.80 = $960
Therefore, the predicted value of the painting in 2018 will be approximately $3,052 + $960 = $4,012.