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Integers Exponents Absolute value

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.

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

Linear Graph

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

variation which is a proportional relationship.

The constant of variation, k, is positive 3.

The line also passes through the origin.

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

variation which is a proportional relationship.

The constant of variation, k, is positive 3.

The line also passes through the origin.

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

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.

is always a vertical line.

Horizontal line:

y=1

The y value is the same for any x value.

is always a horizontal line.

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

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

Nonlinear Graphs \hspace{1.2cm}

Absolute value:

Absolute value graphs are V-shaped

Quadratic:

Quadratic graphs are U-shaped.

Exponential:

Exponential graphs are curves.

Cubic:

Cubic graphs are curves that look like this.

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.

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.

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.

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.

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

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

In order to find the y intercept of a graph:

In order to find the y intercept of an equation:

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

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

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

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Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREE

Use the graph to find the rate of change.

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.

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.

Write 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}{1} \, .

Interpret the slope.

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

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

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

The graph crosses the y axis at 1.

Write the \textbf{y} intercept.

The y intercept is 1.

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 .

\begin{aligned} & y=\cfrac{2}{3} \, x+2 \\\\ & y=\cfrac{2}{3} \, (0)+2 \\\\ & y=0+2 \\\\ & y=2 \end{aligned}

Write the \textbf{y} intercept.

The y intercept is 2.

Interpret the \textbf{y} intercept.

The graph crosses the y axis at positive 2.

Use the graph to find the x intercept.

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

The line crosses the x axis at 2.

Write the \textbf{x} intercept.

The x intercept is 2.

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} .

\begin{aligned} & y=2 x-6 \\\\ & 0=2 x-6 \\\\ & 0+6=2 x-6+6 \\\\ & 6=2 x \\\\ & \cfrac{6}{2}=\cfrac{2 x}{2} \\\\ & 3=x \text { or } x=3 \end{aligned}

Write the \textbf{x} intercept.

The x intercept is 3.

Interpret the \textbf{x} intercept.

The x intercept is where the graph crosses the x axis and is also the root or solution to the equation.

1. Which graph has a y intercept at 7?

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.

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?

The total amount of money she has after 5 years.

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

The initial amount of money she deposited into the account.

The amount of money she made in 5 years.

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} \, ?

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.

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

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

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?

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.

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

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.

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

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.

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.

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