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4. Linear Functions
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United States of AmericaUT
Math I

4.09 Modeling linear relationships

Lesson
Practice
Lesson

Creating linear models

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. 

Slope-intercept form

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=mx+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−y1​=m(x−x1​)

Connections between representations

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.

 

Practice questions

Question 1

Consider the points in the table below:

Time in minutes ($x$x) $1$1 $2$2 $3$3 $4$4 $5$5
Temperature in °C ($y$y) $8$8 $11$11 $14$14 $17$17 $20$20

  1. By how much is the temperature increasing each minute?

  2. What would the temperature have been at time 0?

  3. Which of the following shows the algebraic relationship between $x$x and $y$y?

    $y=5x+3$y=5x+3

    A

    $y=-3x+5$y=−3x+5

    B

    $y=3x+5$y=3x+5

    C

    $y=-5x+3$y=−5x+3

    D

Question 2

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.

  1. Complete the table of values:

    Number of hours passed ($x$x) $0$0 $1$1 $2$2 $3$3 $4$4 $5$5 $10$10
    Amount of water in tank ($y$y) $\editable{}$ $\editable{}$ $36$36 $44$44 $52$52 $\editable{}$ $\editable{}$

  2. Write an algebraic relationship linking the number of hours passed ($x$x) and the amount of water in the tank ($y$y).

  3. Plot the points on the number plane.

    Loading Graph...

 

Interpreting linear models

As mentioned above, we often use slope-intercept form to represent applications of linear relationships. That is we use $y=mx+b$y=mx+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=60x+100 as our equation.

Practice questions

Question 3

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

  1. 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.

  2. What is the slope of the function?

  3. What does this slope represent?

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

    A

    The minimum amount charged by the carpenter.

    B

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

    C

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

    D

  4. What is the value of the $y$y-intercept?

  5. 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.

    A

    The maximum amount charged by the carpenter.

    B

    The callout fee.

    C

    The minimum amount charged by the carpenter.

    D

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

 

Using linear models

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. 

Practice questions

Question 4

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

  1. What is the increase in depth each minute?

  2. Write an equation for the relationship between the number of minutes passed ($x$x) and the depth ($y$y) of the diver.

    Enter each line of work as an equation.

  3. In the equation, $y=1.4x$y=1.4x, what does $1.4$1.4 represent?

    The change in depth per minute.

    A

    The diver’s depth below the surface.

    B

    The number of minutes passed.

    C

  4. At what depth would the diver be after $6$6 minutes?

  5. 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=1.4x.

    Solve this equation for $x$x to find the time it takes.

Question 5

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

Loading Graph...

  1. 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) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

  2. Choose the algebraic equation that represents the relationship between the water temperature ($x$x) and the surface air temperature ($y$y).

    $y=-5x+3$y=−5x+3

    A

    $y=3x+5$y=3x+5

    B

    $y=-3x-5$y=−3x−5

    C

    $y=3x-5$y=3x−5

    D

  3. What would be the surface air temperature when the water temperature is $11$11°C?

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

    $12$12

    A

    $-12$−12

    B

    $88$88

    C

    $-97$−97

    D

Outcomes

I.A.CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 'Linear and exponential (integer inputs only)

I.F.IF.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

I.F.BF.1

Write a function that describes a relationship between two quantities.

I.F.BF.1.a

Determine an explicit expression, a recursive process, or steps for calculation from a context.

I.F.LE.5

Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form f(x) = b^x + k.

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