Interpreting graphs

Here you will learn about interpreting graphs, including how to plot graphs, how to recognize different graphs, the key features of a graph, as well as sketching and transforming graphs.

Students first learn about graphing on the coordinate plane in 6th grade with their work with expressions and equations. They expand this knowledge to graphing linear and non-linear functions in middle school and high school.

What is interpreting graphs?

Interpreting graphs covers the skills of plotting and sketching graphs and identifying the key features of graphs. 

Different types of graphs have specific shapes, and we can recognize the type of graph based on its shape.

Linear graphs

Linear graphs produce a straight line, let’s look at some examples:

Linear Graph
Make a table of values: Select values for x and substitute into the equation to find y. This linear relationship represents direct variation which is a proportional relationship. The constant of variation, k, is positive 3. The line also passes through the origin.
Make a table of values: Select values for x and substitute into the equation to find y. This linear relationship represents direct variation which is a proportional relationship. The constant of variation, k, is positive 3. The line also passes through the origin.
Make a table of values: Select values for x and substitute into the equation to find y. This linear relationship is not direct variation because it does not represent a proportional relationship. The line does not pass through the origin.

Special cases of linear graphs \hspace{1.7cm}

Vertical line: x=1 The x value is the same for any y value. Graphs in the form of x=a, is always a vertical line.	Horizontal line: y=1 The y value is the same for any x value. Graphs in the form of y=a, is always a horizontal line.

You can make a table of values to sketch any type of graph.

Nonlinear graphs

Nonlinear graphs produce graphs in a variety of different shapes, let’s look at some examples:

Nonlinear Graphs \hspace{1.2cm}

Absolute value: y=|x|	Absolute value graphs are V-shaped
Quadratic: y=x^2	Quadratic graphs are U-shaped.
Exponential: y=2^x	Exponential graphs are curves.
Cubic: y=x^3	Cubic graphs are curves that look like this.

Key features of graphs

• \textbf{y} intercept

The y intercept is the point where the graph crosses the y -axis (vertical axis).

For example, the equation y=2x has a y -intercept of 0 because the line passes through the origin, (0, 0).

Another way to find the y intercept is to substitute 0 in for x into the equation and solve for y. The y value is the y intercept.

The line y=x + 4 has a y intercept of 4, meaning it crosses the y axis at 4.

Substitute 0 for x into the equation and solve it for y.

\begin{aligned} & y=x+4 \\\\ & y=0+4 \\\\ & y=4 \end{aligned}

The y -intercept is 4.

• Slope, gradient or rate of change

The slope (also called gradient) is the ratio of the vertical change and horizontal change between two points on a line. It determines how steep the line is.

When the line rises to the right it has a positive slope and when the line rises to the left it has a negative slope.

You can find the gradient of a linear graph by counting the vertical change and the horizontal change from one point to another point on a line and write it as a ratio.

\text { Slope }=\cfrac{\text { Change in } y}{\text { Change in } x}

For example, for the equation y=\cfrac{1}{3} \, x , start at the origin (0, 0) and count up and to the right until you get to the point (3, 1).

You can see that it is 1 unit up and 3 units right. The slope is the ratio of the vertical change over the horizontal change,

*Up is a positive direction.

*Down is a negative direction.

*Right is a positive direction.

*Left is a negative direction.

• \textbf{x} intercepts or roots

The x intercepts or roots of a linear graph is the point where the graph crosses the x axis (horizontal axis).

We can find these points by setting y equal to 0 and then solving the equation.

For example, to find the x intercepts or roots of the graph y=x-5, we substitute 0 for y into the equation to find x.

\begin{aligned} & y=x-5 \\\\ & 0=x-5 \\\\ & 0+5=x-5+5 \\\\ & 5=x \text { or } x=5 \end{aligned}

The x intercept is 5.

• Linear equations can be written in the form \textbf{y = mx} or \textbf{y = mx + b}

m = slope

b= y intercept

y=mx → y intercept of 0 because it represents a proportional linear relationship.

y=mx+b → the value of b is the y intercept.

What is interpreting graphs?

Common Core State Standards

How does this relate to 6th grade math, 7th grade math, and 8th grade math?

• Grade 6 Number System (6.NS.C.6.c)
  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2.b)
  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• Grade 8 – Expressions and Equations (8.EE.B.5)
  Graph proportional relationships, interpreting the unit rate as the slope of the graph.

How to interpret graphs

In order to find and interpret the slope of a linear graph:

1. Identify two points on the graph.
2. Count the vertical movement and horizontal movement from one point to the second point.
3. Write the slope.
4. Interpret the slope.

In order to find the y intercept of a graph:

1. Look for the point where the line crosses the \textbf{y} axis.
2. Write the \textbf{y} intercept.

In order to find the y intercept of an equation:

1. Substitute a \bf{0} for \textbf{x} into the equation and solve it for \textbf{y} .
2. Write the \textbf{y} intercept.

In order to find the x intercept from a linear graph:

1. Look for the point where the line crosses the \textbf{x} axis.
2. Write the \textbf{x} intercept.

In order to find the x intercept from a linear equation:

1. Substitute a \bf{0} for \textbf{y} into the equation and solve it for \textbf{x} .
2. Write the \textbf{x} intercept.
3. Interpret the \textbf{x} intercept.

Interpreting graphs examples

Example 1: find the slope (rate of change)

Use the graph to find the rate of change.

1. Identify two points on the graph.

2Count the vertical movement and horizontal movement from one point to the second point.

3Write the slope.

The slope is the ratio, \text { slope }=\cfrac{\text { change in } y}{\text { change in } x} \, .

So in this case, \text { slope }=\cfrac{1}{2}.

4Interpret the slope.

The slope is a ratio of the vertical change over the horizontal change.

So the slope of \cfrac{1}{2} is the vertical change from the point (0, 0) to (2, 1) , which is 1 unit up, and the horizontal change is 2 units right.

Example 2: find the slope (rate of change)

Use the graph to find the rate of change.

Identify two points on the graph.

Count the vertical movement and horizontal movement from one point to the second point.

Example 3: find the y intercept from a graph

Example 4: find and interpret the y intercept from an equation

What is the y intercept of y=\cfrac{2}{3} x+2 ?

Example 5: find the x intercept from a graph

Use the graph to find the x intercept.

Example 6: find the x intercept from an equation

What is the x intercept of y=2x-6 ?

Teaching tips for interpreting graphs

• Use online interactive graphing tools, graphing apps, or a graphing calculator to help students make connections.

• Although worksheets are good for practicing skills, use real world scenarios to help your learners make connections between the key features of a graph and how they relate to the world around them.

• Reinforce that the slope is a ratio of the vertical change over the horizontal change in two points. It’s essential for long term understanding that the slope is a ratio.

Easy mistakes to make

• Confusing the when to substitute \bf{0}
  For example, substituting 0 in for y when finding the y intercept is incorrect and substituting 0 for x when finding the x intercept is incorrect.

• Mixing up the ratio for slope
  For example, writing the ratio for slope as:
  
  

  

• Confusing the positive or negative slope
  Remember that a positive slope rises to the right and a negative slope rises to the left.

Related coordinate plane lessons

• Coordinate plane
• x and y axis
• Graph transformations
• Plot points on a graph
• Independent and dependent variables

The next lessons are

• Types of graphs
• Graphing linear equations
• Rate of change

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs.