**Included topics:**linear functions; linear equations; systems of linear equations; linear inequalities.

**Included topics:**quadratic equations and functions; graphing quadratic functions; linear and quadratic systems of equations; exponential expressions; graphing exponential functions; polynomial functions and graphs; radical and rational equations and expressions; isolating quantities; function notation.

**Included topics:**ratios rates and proportions; percentages; units and unit conversion; linear and exponential growth; table data; data collection and data inference; center spread and shape of distributions; key features of graphs; scatterplots.

**Included topics:**complex numbers; volume word problems; congruence and similarity; right triangle trigonometry and word problems; circle theorems; angles, arc lengths and trig functions; circle equations.

# Key features of graphs on the SAT test

## SAT Subscore: Problem solving and data analysis

### Studying key features of graphs

**On the SAT test key features of graphs topic** is part of problem solving and data analysis subscore that includes 9 advanced topics (see the full topics list on the top menu).

**Key features of graphs topic is the eighth topic** of problem solving and data analysis subscore. It is recommended to start learning problem solving and data analysis subscore with its first topic called ratios, rates and proportions.

**Key features of graphs topic is divided into sections** from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

**Finish studying** heart of algebra subscore topics before you study this topic or any other problem solving and data analysis subscore topic. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding problem solving and data analysis subscore topics).

### Key features of graphs- summary

**A graph is defined as** a pictorial representation of data or numeric values in an organized manner. Using graph enables us to represent large amounts of data in visual form for easy understanding.

**The 2 types of questions about the key features of graphs** are interpreting given graphs or selecting a graph based on a verbal description.

**The common graph types** on SAT are bar graphs, dot plots, histograms, line graphs and scatterplots. Scatterplot subject is covered on scatterplots page.

**A Bar Graph** is a graphical display of data using rectangular bars (columns) of different heights, so that the height of each bar determines its value. The structure of the bar graph: the y axis contains values, and the x axis contains categories or time periods.

**A dot plot** is a simple type of graph that shows the frequency with which items appears in a data set. It displays data items as dots above values (or categories) on the x axis (each data item is represented with a dot above its value or category).

**A histogram** is a frequency bar graph where the data is grouped into ranges. The x axis presents the data ranges and the y axis presents the frequency (the number of values that fall into the specific range.

**A line graph** includes a line that connects individual data points together. Line graphs are used to show changes over periods of time, so that the x axis represents time values (like years).

**Continue reading this page for detailed explanations and examples.**

### Bar graphs

**A Bar Graph** is a graphical display of data using rectangular bars (columns) of different heights, so that the height of each bar determines its value (the larger the value the higher the bar).

**A bar graph purpose** is to compare values between different groups or to compare values over time.

**A bar graph structure:** the y axis contains values, and the x axis contains variables of 2 types: group types (categories) or time periods.

#### Bar graphs that compare values over time

The following bar graph shows a quarterly revenue trend of company A by comparing __values (revenues) over time ( 4 quarters) __(in dollars). The data that was used for the bar graph is presented in the table below the graph.

The bar graph above shows that:

- The revenue is rising every quarter (the bar of each quarter is higher than the bar of the previous quarter).
- The lowest quarterly revenue is 70,000 dollars and the highest quarterly revenue is 140,000 dollars.
- The revenue increase in the third quarter (110,000-80,000 dollars) equals to the revenue increase in the fourth quarter (140,000-10,000 dollars).
- The revenue increase in the third quarter (110,000-80,000 dollars) is 3 times bigger than the revenue increase in the second quarter (80,000-70,000 dollars).

#### Bar graphs that compare values between different groups

The following bar graph shows the revenue of 3 companies in the first quarter. The graph compares __values (revenues) between different groups (companies) __(in dollars). The data that was used for the bar graph is presented in the table below the graph.

The bar graph above shows that:

- The company B has the highest revenue in the first quarter (140,000 dollars).
- The company C has the lowest revenue in the first quarter (10,000 dollars).
- The revenue of company B (140,000 dollars) is twice bigger than the revenue of company A (70,000 dollars).
- The difference between the revenue of company A (70,000 dollars) and the revenue of company C (10,000 dollars) is 60,000 dollars.

#### Bar graphs that compare values between different groups over time

We can combine the 2 graphs from above into one graph by __showing different groups over different time periods__. To do so we need to show the data of the 3 groups (companies) in every quarter. Each company data bar is represented in different color to separate the data of that company from the other companies.

The following bar graph shows the revenue of 3 companies in 4 quarters. The graph compares __values (revenues) between different groups (companies) over time (quarters) __(in dollars). The data that was used for the bar graph is presented in the table below the graph.

The bar graph above shows that:

- The revenue of company A is rising every quarter (the bar of each quarter is higher than the bar of the previous quarter).
- The revenue of company B is declining every quarter (the bar of each quarter is lower than the bar of the previous quarter).
- The revenues of company A and company B are always higher than the revenues of company C.
- The revenue of company B in the first quarter is equal to the revenue of company A in the fourth quarter (140,000 dollars).

#### Stacked columns bar graphs

The following stacked bar graph shows the revenue of 3 companies in 4 quarters. In this bar graph the data series are stacked one on top of the other in vertical columns. This presentation method allows us to see the total revenue value in each quarter and perform part to whole comparisons.

**Note that** finding revenue values of companies B and C that are stacked above company A requires calculating a difference between two values. See example below.

The bar graph above shows that:

- The total revenue of the 3 companies was the highest in the second quarter (225,000 dollars).
- The revenue of company B in the third quarter is 90,000 dollars (200,000-110,000 dollars).
- The revenue of company C is approximately 5 percent of the total revenue in the fourth quarter (10,000/90,000 dollars= 1/19*100%=5%).

### Dot plots

**A dot plot** is a simple type of graph that shows the frequency with which items appears in a data set. It displays data items as dots above values (or categories) on the x axis (each data item is represented with a dot above its value or category).

**How to draw a dot plot:**

Step 1- Arrange the data values (categories) in increasing order.

Step 2- Count the number of values (categories) for each grade.

Step 3- Write the values (categories names) on the x axis.

Step 4- Draw dots above each value (category).

Consider the following example:

The exam grades of 10 students are: 80, 90, 70, 65, 90, 75, 65, 80, 90 and 95.

Which pot plot describes these grades?

**Step 1- Arranging the data:**

Before drawing the dot plot we need to arrange the grades in increasing order: 65, 65, 70, 75, 80, 80, 90, 90, 90 and 95.

**Step 2- Counting the number of students for each grade:**

65-2, 70-1, 75-1, 80-2, 90-3, 95.

**Steps 3 and 4- drawing the graph:**

The x axis should present values (the grades) and each grade should be represented by a dot above the grade value. For example: since 3 students received a grade of 90 there should be 3 dots above the grade 90.

The following dot plot represents the given data.

The dot plot above shows that:

- No student received a grade of 85.
- The number of students that received a grade of 70 is equal to the number of students that received a grade of 75 (1 student).
- The number of students that received the grade 65 is twice bigger than the number of students that received the grade 70.

### Histograms

**A histogram** is a frequency bar graph where the data is grouped into ranges. The x axis presents the data intervals (ranges) and the y axis presents the frequency (the number of values that fall into the specific interval/ range). Taller bar shows that more data falls into the range of the bar.

**Note that:**

- All bars have the same width.
- All bars touch each other, because the scale is continuous (bars in the regular bar graph don’t touch each other).
- The histogram does not give any information about the specific sizes of the values, it only shows how many values are in any data range/ interval.

**Drawing a histogram steps:**

Step 1: Arrange the data values in increasing order.

Step 2: Determine the size of the range.

Step 3: Place the ranges on the x axis.

Step 4: Place the frequencies on the y axis.

Step 5: Draw a bar in each interval. The height of each bar is equal to its frequency.

Consider the previous example:

The exam grades of 10 students are: 80, 90, 70, 65, 90, 75, 65, 80, 90 and 95.

Which histogram describes these grades?

**Step 1:** Arranging the data values in increasing order.

Before drawing the histogram, we need to arrange the grades in increasing order: 65, 65, 70, 75, 80, 80, 90, 90, 90 and 95.

**Step 2:** Determining the size of the range.

The size of the range that we choose is 5 points, and the ranges are 65-70, 70-75, 75-80, 80-85, 85-90, 90-95.

The following histogram represents the given data.

**Note that** the histogram does not give any information about the specific values of the grades, it only shows how many grades (students) are in any grade range/ interval. We know that 1 student received a grade between 70-75, but we do not know from the histogram what was the grade value.

**Note that** the value 70 that appears in the first bar and the second bar is counted in the first bar (the first bar from the two bars), the value 90 that appears in the fifth and sixth bars is counted in the fifth bar (the first bar from the two bars) etc.

The histogram above shows that:

- No students received grades between 80-85.
- The number of students that received grades between 70-75 is equal to the number of students that received grades between 90-95 (1 student).
- The number of students that received grades between 65-70 is 3 times bigger than the number of students that received grades between 70-75.

### Line graphs

**A line graph** includes a line that connects individual data points together. Line graphs are used to show changes over periods of time, so that the x axis represents time values (like years).

**Line graphs question types:**

- Identifying values on the graph.
- Analyzing the direction of the change in the different areas of the graph (increasing or decreasing).
- Finding the rate of the change between different points of the graph (done by analyzing at the steepness of the graph).

__Drawing a line graph steps:__

Step 1: Write the names of the axis (the x axis should represent time values).

Step 2: Plot the data points on the graph.

Step 3: Draw a line from each point to the point near it.

The following line graph shows the number of students that graduated from high school A from the year 2010 to 2020, the table that includes the data for the line graph is located below the histogram.

The line graph above shows that:

- The number of students that graduated from high school A increased from the year 2012 to 2020.
- The lowest number of students that graduated from high school A was in the year 2010.
- The highest change in the number of students that graduated from high school A was between the years 2012 and 2013.

### Finding a graph that represents a word problem

In these questions, we need to transfer a word problem to graphical terms. Below is an explanation for finding a line graph, graphs of other types can be built in a similar way.

**Finding a line graph that represents a word problem steps:****Step 1:** Find the x and y axes names- Find the axis names in the word problem question. Analyze the question to determine which variable is dependent (should appear on the y axis) and which is independent (should appear on the x axis).

Note that time is an independent variable and must be on the x axis.**Step 2: **Divide the word question into parts (in each part the y value changes therefore the shape of the graph changes).**Step 3:** Calculate the x and y values for each part of the word question.**Step 4:** Draw the graph for each part of the word question identified in step 2 according to the x and y values calculated in step 3- The shape of the graph should be according to the information given in the word problem.

**The properties of the shape of the graph:**

The trend (direction) of the graph– the graph can go up (upward trend), go down (downward trend) or remain the same (flat trend).

The slope of the graph- the graph can change fast (steep slope) or slow (shallow slope).

Consider the following example:

The factory worker operates a machine that produces chocolate bars. He counts the number of the chocolate bars every hour and packs them in a box at the end of the day. In the morning, the worker saw that yesterday he forgot to put 5 bars in the box and turned the machine on. He decided to add the 5 bars to the bars produced today. After the first hour the worker counted 50 bars. After the second hour the worker calculated 20 percent less bars than after the first hour. Then he took a break for an hour. After he returned the worker operated the machine for the last hour (the fourth hour) and counted a total of 140 bars.

Draw a graph that represents the number of the chocolate bars produced as time passed.

**Step 1: Finding the x and y axes names:**

The question asks about the number of the chocolate bars produced as time passes, therefore the axis names are the number of the chocolate bars produced and the time (in hours). Since the number of the chocolate bars produced changes as time passes, the independent variable is the time (should appear on the x axis) and the dependent variable is the number of the chocolate bars produced (should appear on the y axis).

**Step 2:** **Dividing the word question into parts:**

Each hour the number of chocolate bar changes, therefore we need to divide the word question into 4 parts.

**Step 3:** **Calculating the x and y values for each part of the word question:**

Starting point x=0: In the morning there were 5 chocolate bars, therefore the point is (0,5).

After 1 hour x=1: In the first hour the machine produced 50 bars, therefore the point is (1,55).

After 2 hours x=2: In the second hour the machine produced 20 percent less bars than in the first hour, therefore the machine produced 50*(100%-20%)=50*80%=50*80/100=40 bars, therefore the point is (2,95).

After 3 hours x=3: In the third hour the machine didn’t work, therefore the point is (3,95).

After 4 hours x=4: After four hours the worker counted a total of 140 bars, therefore the point is (4,140).

The following line graph shows the number of the chocolate bars produced as time passed.

**scatterplots.**