4. Linear Functions

Lesson

Now that we know how

- to graph linear relationships
- to find the equations of linear functions
- to use algebra and graphs to extract information
- to find intercepts and constant values, and
- that the slope of a linear function represents constant change.

We can put this to use to solve a range of real life applications.

It's all the same mathematics, but this time you will have a context to apply it to.

When given an equation in slope-intercept form, we can interpret the values of the slope and the $y$`y`-intercept of a line.

When given a scenario or a table of values, we often will try to write a linear equation in slope-intercept form.

$y=mx+b$`y`=`m``x`+`b`

$m$`m` is the slope and $b$`b` is the $y$`y`-intercept

$m$`m` is the unit rate of change, so as $x$`x` increases by one unit, $y$`y` will increase/decrease by $m$`m` units. Sometimes this will be given to us, other times we will need to calculate it from two or more points.

$b$`b` is the initial value, so when $x=0$`x`=0, $y=b$`y`=`b`. Again, this might be given, or we might need to do some calculations to find it.

We may also use point-slope form to get the equation as well. Remember that is $y-y_1=m\left(x-x_1\right)$`y`−`y`1=`m`(`x`−`x`1)

We can represent a linear relationship in many different ways including tables of values, graphs, or equations. This table below show how the slope and y-intercept can be seen from all representations.

Slope | y-intercept | |
---|---|---|

Table of values | The constant rate of change. If the x-value is increasing by $1$1, then it will be the increase in the y-values. | The value of $y$y when $x=0$x=0. This may not actually be given in the table, so you may have to count backwards using the rate of change. |

Graph | The slope can be read using the rise and run between two points. | The value where the linear graph crosses the $y$y-axis. |

Equation | In slope-intercept form, it will be the coefficient of the x-value. $m$m in $y=mx+b$y=mx+b. |
In slope-intercept form, it will be the constant. $b$b in $y=mx+b$y=mx+b. |

Consider the points in the table:

Time in minutes ($x$x) |
1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Temperature in °C ($y$y) |
$6$6 | $9$9 | $12$12 | $15$15 | $18$18 |

By how much is the temperature increasing each minute?

What would the temperature have been at time 0?

Find the algebraic relationship between $x$

`x`and $y$`y`.

There are $20$20 liters of water in a rainwater tank. It rains for a period of 24 hours and during this time, the tank fills up at a rate of $8$8 Liters per hour.

Complete the table of values:

Number of hours passed ($x$ `x`)$0$0 $1$1 $2$2 $3$3 $4$4 $4.5$4.5 $10$10 Amount of water in tank ($y$ `y`)$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Write an algebraic relationship linking the number of hours passed ($x$

`x`) and the amount of water in the tank ($y$`y`).Plot the points on the number plane.

Loading Graph...

As mentioned above, we often use slope-intercept form to represent applications of linear relationships. That is we use $y=mx+b$`y`=`m``x`+`b` where $m$`m` is the slope or rate of change and $b$`b` is the $y$`y`-intercept or initial value.

A common scenario is a cost function where there is both a fixed and variable component. For example, a plumber that charges a flat-fee of $\$100$$100 to show up plus an hourly rate of $\$60$$60. The initial or flat-fee would be the $y$`y`-intercept, while the rate would be the slope. We would get $y=60x+100$`y`=60`x`+100 as our equation.

A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.

Write an equation to represent the total amount charged, $y$

`y`, by the carpenter as a function of the number of hours worked, $x$`x`.What is the slope of the function?

What does this slope represent?

The total amount charged increases by $\$45$$45 for each additional hour of work.

AThe minimum amount charged by the carpenter.

BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.

CThe total amount charged for $0$0 hours of work.

DThe total amount charged increases by $\$45$$45 for each additional hour of work.

AThe minimum amount charged by the carpenter.

BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.

CThe total amount charged for $0$0 hours of work.

DWhat is the value of the $y$

`y`-intercept?What does this $y$

`y`-intercept represent?Select all that apply.

The total amount charged increases by $\$150$$150 for each additional hour of work.

AThe maximum amount charged by the carpenter.

BThe callout fee.

CThe minimum amount charged by the carpenter.

DThe total amount charged increases by $\$150$$150 for each additional hour of work.

AThe maximum amount charged by the carpenter.

BThe callout fee.

CThe minimum amount charged by the carpenter.

DFind the total amount charged by the carpenter for $6$6 hours of work.

Once we have a linear model, we can use it to make predictions. If we substitute in for one variable we can use the linear equation to solve or evaluate for the other.

Careful!

When you are substituting into a linear equation, be sure that you are substituting for the correct variable.

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver over $5$5 minutes.

Number of minutes passed ($x$x) |
$0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|

Depth of diver in meters ($y$y) |
$0$0 | $1.4$1.4 | $2.8$2.8 | $4.2$4.2 | $5.6$5.6 |

What is the increase in depth each minute?

Write an equation for the relationship between the number of minutes passed ($x$

`x`) and the depth ($y$`y`) of the diver.In the equation, $y=1.4x$

`y`=1.4`x`, what does $1.4$1.4 represent?The change in depth per minute.

AThe diver’s depth below the surface.

BThe number of minutes passed.

CThe change in depth per minute.

AThe diver’s depth below the surface.

BThe number of minutes passed.

CAt what depth would the diver be after $6$6 minutes?

We want to know how long the diver takes to reach $12.6$12.6 meters beneath the surface.

If we substitute $y=12.6$

`y`=12.6 into the equation in part (b) we get $12.6=1.4x$12.6=1.4`x`.Solve this equation for $x$

`x`to find the time it takes.

The points show the relationship between water temperatures and surface air temperatures.

Loading Graph...

Complete the table of values:

Water Temperature (°C) $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 Surface Air Temperature (°C) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Write an algebraic equation representing the relationship between the water temperature ($x$

`x`) and the surface air temperature ($y$`y`).What would be the surface air temperature when the water temperature is $11$11°C?

What would be the water temperature when the surface air temperature is $31$31°C?

Write each line of work as an equation.

Use the concept and notation of functions to interpret and apply them in terms of their context.

Write functions or sequences that model relationships between two quantities.

Create and graph equations or inequalities to describe numbers or relationships.