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3. Relations & Functions
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Grade 8

3.05 Characteristics of linear functions

Lesson
Practice

Characteristics of linear functions

Linear functions can be represented multiple ways, with each representation displaying key characteristics of the function. We can use the type of representation the function to determine key points and the pattern of how the x-values and y-values change.

The table below represents the recorded growth of a plant over the course of several weeks.

\text{weeks} \, (x)02468
\text{growth in inches} \, (y)34567

When this set of ordered pairs is graphed on a coordinate plane, we see it forms a straight line. Drawing this line can help us predict the growth of the plant at weeks other than the ones included in the initial table.

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\text{weeks}
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\text{growth (in)}

Recall that slope is a number that represents the steepness of a line, its sign determines how the line rises or falls from left to right.

Slope

The rate of change in a linear function or the “steepness” of the line. The slope of a line is a rate of change, a ratio describing the vertical \left(y\right) change to the horizontal \left(x\right) change.

There are four types of slope:

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Zero
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Undefined

The slope of the line describing the plant's growth is positive, so we know that the plant's height is increasing as time passes. The specific values in the table and graph can be used to find the growth rate of the plant.

table of values with the rows labeled as weeks (x) and growth in inches (y) for the points (0,3),(2,4),(4,5),(6,6), and (8,7) showing the rate of change for the y-values as +1 and the x-values as +2.

From the table we see that as each y-value increases by 1, the x-values increase by 2. This means that the plant grows 1 inch every 2 weeks.

Slope is written as the ratio: \dfrac{\text{change in } y}{\text{change in }x}

The ratio, and therefore the slope, for this table is \dfrac{+1}{+2}=\dfrac{1}{2}.

On a coordinate plane, we can visualize the slope with triangles to show the vertical and horizontal change between points.

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\text{weeks}
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\text{growth (in)}

The smaller triangles on the graph show a vertical change of +1 square and a horizontal change of +2 squares. Giving a ratio of \dfrac{1}{2}.

The larger triangle on the graph shows a vertical change of +3 squares and a horizontal change of +6 squares. This simplifies to \dfrac{1}{2}, the same slope as seen from the smaller triangles and the original table.

The slope is not the only value that provides information about a linear function given a table or graph. Knowing where an x-value or y-value equals 0 gives context for the situation described.

y-intercept

The point at which the graph of the function intersects the y-axis and may be given as a single value, b, or as the location of a point \left(0, b\right).

The y-intercept represents an initial (starting) value for the dependent variable, y.

In a table, the entry where the x-value is 0 represents the y-intercept. In this table, the y-intercept means that the plant measured 3 inches at the time we started collecting data. The number of weeks in the table are counting starting from this point.

table of values with the rows labeled as weeks (x) and growth in inches (y) for the points (0,3),(2,4),(4,5),(6,6), and (8,7). (0,3) is encircled.
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\text{weeks}
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\text{growth (in)}

On a graph, the y-intercept is found where the function crosses the y-axis, or the point \left(0,y\right).

This line crosses the y-axis at \left(0,3\right). And, just like in the table, we can interpret this point to mean "at 0 weeks, the plant was 3 inches tall."

Other points on the line can be described in a similar way. For example, the point \left(4,5\right) can be interpreted as "the plant measured 5 inches tall 4 weeks after the initial measurement."

Examples

Example 1

Consider the equation y=3x+1.

a

Complete the table of values shown:

x-1012
y
Worked Solution
Create a strategy

Substitute each values from the tables into the given equation.

Apply the idea

For x=-1:

\displaystyle y\displaystyle =\displaystyle 3\cdot \left(-1\right)+1Substitute -1 for x
\displaystyle =\displaystyle -2Evaluate

Similarly, if we substitute the other values of x, \left( x=0,\, x=1,\, x=2\right), into y=3x+1, we get:

x-1012
y-2 14 7
b

Plot the points in the table of values.

Worked Solution
Create a strategy

For an ordered pair \left(a,b\right) from the given table of values found in part (a), identify where x=a along the x-axis and y=b along the y-axis.

Apply the idea

Since we are given table of values, then the ordered pairs of points to be plotted on the coordinate plane are \left(-1,\,-2\right),\, \left(0,\,1\right),\, \left(1,\,4\right) and \left(2,\,7\right).

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c

Draw the graph of y=3x+1.

Worked Solution
Create a strategy

Use the plotted points on the coordinate plane from part (b).

Apply the idea

The equation y=3x+1 must pass through each of the plotted points.

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Reflect and check

Both the graph and the table from part (a) show the y-intercept at \left(0,1\right). If 0 had not been included in the table the graph or equation y=3x+1 could be used to determine the y-intercept.

Example 2

The graph of a linear function is shown.

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a

Determine the slope.

Worked Solution
Create a strategy

Find points graphed on the line and draw slope triangles using these points. The slope is the ratio of \dfrac{\text{vertical change}}{\text{horizontal change}} .

Apply the idea

We will use slope triangles to help determine the vertical and horizontal change.

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To get from one point to the next point further to the right we move down 2 units and right 3 units.

The ratio of vertical change to horizontal change is \dfrac{-2}{3} or -\dfrac{2}{3}.

The slope is -\dfrac{2}{3}.

Reflect and check

Making a table of values from the graph can also be used to find the slope.

x-6-303
y1-1-3-5

From the table, we see the y-values change by -2 and the x-values change by +3. The slope is the ratio of the change in y to the change in x, so the slope is -\dfrac{2}{3}.

b

Identify the y-intercept.

Worked Solution
Create a strategy

The y-intercept is the point \left(0,y\right) where the line crosses the y-axis.

Apply the idea

The line crosses the y-axis at the point point \left(0,-3\right).

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The y-intercept is \left(0,-3\right).

Reflect and check

The y-intercept is found in a table by finding the y-value that goes with the x-value of 0.

x-6-303
y1-1-3-5

The y-value that goes with the x-value of 0 is -3 so the y-intercept is -3 or written as an ordered pair \left(0, -3\right).

Example 3

Consider the table of values.

x-1012
y-5-214
a

Identify the coordinates of the y-intercept.

Worked Solution
Create a strategy

We can use the fact that the y-intercept is the point where the x-coordinate is always 0.

Apply the idea

Based on the table, when the value of x is 0, the value of y is -2. This means that the y-intercept is \left(0,-2\right).

Reflect and check

By graphing the points from the table, the y-intercept is represented by a point on the y-axis. The pair \left(0,-2\right) from the original table shows as a point on the y-axis, which makes it the y-intercept.

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b

Determine the slope.

Worked Solution
Create a strategy

Find the change in y-values, change in x-values, then write as the ratio of \dfrac{\text{change in }y}{\text{change in }x}.

Apply the idea

The y-values \left\{-5,-2,1,4\right\} change by +3.

The x-values \left\{-1,0,1,2\right\} change by +1.

The ratio of the change in y to the change in x is \dfrac{3}{1}. This ratio should be simplified as a reduced improper fraction or integer if possible before writing as slope. The slope is 3.

Reflect and check

The slope can also be found on a graph by writing and simplifying the ratio of vertical change to horizontal change between points from left to right.

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The vertical change is +3, the horizontal change is +1, so the ratio would be \dfrac{3}{1}. This would simplify to a slope of 3.

Example 4

On a summer day, the temperature starts at 80 \degree in the morning and rises 2 \degree every hour for several hours. Use this information to complete the table of values.

\text{Hours }(x)02468
\text{Temperature }(y)
Worked Solution
Create a strategy

Use the initial value of x and y with the unit rate to determine missing values. The table's x-values do not change by 1, so the change in y will need to be adjusted to match the change in x.

Apply the idea

The initial value of the table will be \left(0,80\right) because the starting temperature is 80 \degree.

Notice we are told the temperature rises by 2 \degree every 1 hour but the table is numbered in 2 hour increments. To find the temperature at 2 hours we will start a 80 \degree and increase by 2 \degree twice.

\displaystyle 80\displaystyle =\displaystyle 80+00 hours
\displaystyle 82\displaystyle =\displaystyle 80+21 hour
\displaystyle 84\displaystyle =\displaystyle 80+2+22 hours

We see that for every 2 hours, the temperature increases by 4 \degree. To complete the table, each previous y-value will be increased by 4.

\displaystyle 80\displaystyle =\displaystyle 80+00 hours
\displaystyle 84\displaystyle =\displaystyle 80+42 hours
\displaystyle 88\displaystyle =\displaystyle 84+44 hours
\displaystyle 92\displaystyle =\displaystyle 88+46 hours
\displaystyle 96\displaystyle =\displaystyle 92+48 hours
\text{Hours }(x)02468
\text{Temperature }(y)8084889296
Idea summary

Linear functions can be represented as equations, tables of values and graphs.

The slope of a linear function represents

\dfrac{\text{change in }y}{\text{change in }x} = \dfrac{\text{vertical change}}{\text{horizontal change}}

The y-intercept of a linear function in a table is the ordered pair \left(0,y\right) and the point on a graph where the function crosses the y-axis.

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Outcomes

8.PFA.3

The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

8.PFA.3b

Describe key characteristics of linear functions including slope (m), y-intercept (b), and independent and dependent variables.

8.PFA.3c

Graph a linear function given a table, equation, or a situation in context.

8.PFA.3d

Create a table of values for a linear function given a graph, equation in the form of y = mx + b, or context.

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