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) | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|

\text{growth in inches} \, (y) | 3 | 4 | 5 | 6 | 7 |

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.

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.

There are four types of slope:

Positive

Negative

Zero

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.

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.

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.

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.

Consider the equation y=3x+1.

a

Complete the table of values shown:

x | -1 | 0 | 1 | 2 |
---|---|---|---|---|

y |

Worked Solution

b

Plot the points in the table of values.

Worked Solution

c

Draw the graph of y=3x+1.

Worked Solution

The graph of a linear function is shown.

a

Determine the slope.

Worked Solution

b

Identify the y-intercept.

Worked Solution

Consider the table of values.

x | -1 | 0 | 1 | 2 |
---|---|---|---|---|

y | -5 | -2 | 1 | 4 |

a

Identify the coordinates of the y-intercept.

Worked Solution

b

Determine the slope.

Worked Solution

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) | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|

\text{Temperature }(y) |

Worked Solution

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.