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

Interpreting graphs y=3 x
Make a table of values:

Select values for x and substitute into the equation to find y.

Interpreting graphs group table image 1
Interpreting graphs group table image 2
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.


Interpreting Graphs y=-3 x
Make a table of values:

Select values for x and substitute into the equation to find y.

Interpreting graphs group table image 1
Interpreting graphs group table image 2
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.


Interpreting Graphs y=3 x-2
Make a table of values:

Select values for x and substitute into the equation to find y.

Interpreting graphs group table image 1
Interpreting graphs group table image 2
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.

Interpreting graphs group table image 10
Interpreting graphs group table image 11
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.

Interpreting graphs group table image 12
Interpreting graphs group table image 13
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|
























Interpreting graphs example icon 1
Interpreting graphs group table image 14
Interpreting graphs group table image 15

Absolute value graphs are V-shaped

Quadratic:
y=x^2
























Interpreting graphs example icon 1
Interpreting graphs group table image 16
Interpreting graphs group table image 17

Quadratic graphs are U-shaped.

Exponential:
y=2^x






























Interpreting graphs example icon 1
Interpreting graphs group table image 18
Interpreting graphs group table image 19

Exponential graphs are curves.

Cubic:
y=x^3




























Interpreting graphs example icon 1
Interpreting graphs group table image 14
Interpreting graphs group table image 21

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).

Interpreting Graphs image 4

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.

Interpreting Graphs image 5

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,

Interpreting Graphs image 6

*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 β†’ y intercept of 0 because it represents a proportional linear relationship.

y=mx+b β†’ the value of b is the y intercept.

What is interpreting graphs?

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.

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[FREE] Algebra Check for Understanding Quiz (Grade 6 to 8)

[FREE] Algebra Check for Understanding Quiz (Grade 6 to 8)

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Use this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!

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Interpreting graphs examples

Example 1: find the slope (rate of change)

Use the graph to find the rate of change.

Interpreting Graphs example 1 image 1

  1. Identify two points on the graph.

Interpreting Graphs example 1 image 2

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

Interpreting Graphs example 1 image 3

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.

Interpreting Graphs example 2 image 1

Identify two points on the graph.

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

Write the slope.

Interpret the slope.

Example 3: find the y intercept from a graph

Interpreting Graphs example 3

Look for the point where the graph crosses the \textbf{y} axis.

Write the \textbf{y} intercept.

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

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

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

Write the \textbf{y} intercept.

Interpret the \textbf{y} intercept.

Example 5: find the x intercept from a graph

Use the graph to find the x intercept.

Interpreting Graphs example 5

Look for the point where the line crosses the \textbf{x} axis.

Write the \textbf{x} intercept.

Example 6: find the x intercept from an equation

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

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

Write the \textbf{x} intercept.

Interpret the \textbf{x} intercept.

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:

    Interpreting Graphs image 7

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

Practice interpreting graphs questions

1. Which graph has a y intercept at 7?

Interpreting Graphs practice question 1 image 1

GCSE Quiz True

Interpreting Graphs practice question 1 image 2

GCSE Quiz False

Interpreting Graphs practice question 1 image 3

GCSE Quiz False

Interpreting Graphs practice question 1 image 4

GCSE Quiz False

Look to see which of the lines cross the y axis at positive 7.

 

Where the graph (in this case line) crosses the y axis is the y intercept.

 

Interpreting Graphs practice question 1 image 5

2. Maddie deposits \$100 in a savings account. The graph shows how much money she makes over 5 years. What does the y intercept represent?

 

Interpreting Graphs practice question 2

The total amount of money she has after 5 years.

GCSE Quiz False

The rate of the amount of money she makes each year.

GCSE Quiz False

The initial amount of money she deposited into the account.

GCSE Quiz True

The amount of money she made in 5 years.

GCSE Quiz False

From the graph you can see that the y intercept is 100.

 

\$100 is the initial amount of money Maddie deposited into the savings account, so the y intercept represents the initial amount of money.

3. Which graph has a slope of \, \cfrac{2}{5} \, ?

Interpreting Graphs practice question 3 image 1

GCSE Quiz False

Interpreting Graphs practice question 3 image 2

GCSE Quiz True

Interpreting Graphs practice question 3 image 3

GCSE Quiz False

Interpreting Graphs practice question 3 image 4

GCSE Quiz False

The slope is the ratio of the vertical change over the horizontal change between two points.

 

Identify two points and count the vertical movement and horizontal movement from one point to the next point.

 

Interpreting Graphs practice question 3 image 5

 

In this case, the \text{slope } = \cfrac{2}{5}

4. What is the x intercept of y = 3x – 9?

9
GCSE Quiz False

-3
GCSE Quiz False

-9
GCSE Quiz False

3
GCSE Quiz True

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

 

\begin{aligned} & y=3 x-9 \\\\ & 0=3 x-9 \\\\ & 0+9=3 x-9+9 \\\\ & 9=3 x \\\\ & \cfrac{9}{3}=\cfrac{3 x}{3} \\\\ & 3=x \end{aligned}

 

The x intercept is 3.

5. What is the slope of this linear graph?

 

Interpreting Graphs practice question 5 image 1

-\cfrac{1}{3}
GCSE Quiz False

-\cfrac{3}{1}
GCSE Quiz False

\cfrac{3}{1}
GCSE Quiz True

\cfrac{1}{3}
GCSE Quiz False

The slope is the ratio of the vertical change over the horizontal change between two points.

 

Identify two points and count the vertical movement and horizontal movement from one point to the next point.

 

Interpreting Graphs practice question 5 image 3

6. What is the y intercept of y=3x-4?

-4
GCSE Quiz True

4
GCSE Quiz False

3
GCSE Quiz False

-3
GCSE Quiz False

Substitute 0 in for x and solve for y.

 

\begin{aligned} & y=3 x-4 \\\\ & y=3(0)-4 \\\\ & y=-4 \end{aligned}

 

The y intercept is -4.

 

You can also compare the equation to y=mx+b to determine the value of b.

 

In this case, b=-4 so the y intercept is -4.

Interpreting graphs FAQs

Can a graph have an \textbf{x} -intercept and not a \textbf{y} -intercept?

Yes, think about a vertical line. It has an x -intercept but not a y -intercept.

What is function notation?

Function notation is represented by f(x) which can be substituted for y. Equations can be rewritten in function notation. You will learn more about function notation and graphs of functions in algebra.

Does the slope have to be a fraction?

The slope of a line is a ratio. Ratios are typically written as a fraction but it can also be represented by a decimal number.

The next lessons are

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