Here are a few recommended readings before getting started with this lesson.

Mark always likes to have cans of his favorite soft drink in the refrigerator so that he can enjoy a cold drink whenever he wants.

At the beginning of the week, he had cans in the fridge. Over the week, Mark wrote down each time cans were consumed or added.

A piecewise function is shown below. For every given value of find the correct value of Round the result to two decimal places, if necessary.

When graphing a piecewise function, each piece must be considered separately. First, a piece is graphed for the values of its domain. Then an end — or ends — of the piece are marked with one of the following.

The first piece is This is a linear function written in slope-intercept form. The function can be graphed by using the intercept and the slope.

Now, since the function is limited to inputs less than or equal to the line will be graphed until it reaches Since the inequality is non-strict, the function is defined for and the circle will be closed.

A similar process can be repeated to graph the second piece First, the line will be graphed.

This piece is defined for values of greater than This means that the line will be drawn starting at Also, since the inequality is strict, the circle will be open.

Finally, the pieces will be graphed together in the same coordinate plane to complete the graph. It is important to pay attention to the limit points, since each value of must be assigned to only one value of the function.

The second piece of the function is the horizontal line Also, since the inequality is non-strict, the circle at will be filled.

Now, the intercept can be used to graph the third and final piece. From Part A, it is known that the intercept is

Finally, the graph of the piecewise function is completed.

As a summer activity, Tearrik participates in charity events for his community. He is volunteering for a food drive event this weekend.

He went to a shopping center multiple times over the week to collect boxes of food for the food drive. On each day, he collected the following number of boxes.

Day | Collected Boxes | Boxes in Total |

Sunday | – | |

Monday | ||

Tuesday | ||

Wednesday | ||

Thursday | ||

Friday |

Since step functions are piecewise functions, to graph them, each piece must be considered separately. First, each piece is graphed in its domain as a part of a horizontal line. The end or ends are then marked with one of the following.

The first piece is defined over the interval This piece will be graphed by drawing a horizontal line at from to

It can be noted that is included in the domain of the first piece, but is not. Therefore, the left end of the segment will be marked with a closed circle and the right end with an open circle.

Then, the same process is repeated for each piece of the function.

Zain is working as a server in a restaurant for a week before their summer vacation ends.

Most of their payment comes from the tips they receive. Zain made note of how much they received in tips, starting from Monday and through their last day working on Saturday.

Day | Tips Received | Total Tips |

Sunday | – | |

Monday | ||

Tuesday | ||

Wednesday | ||

Thursday | ||

Friday | ||

Saturday |

The next piece is the horizontal segment at with a closed endpoint at and an open endpoint at

The remaining pieces can be added to the graph by following the same process.

Considering more values can help understand how to draw the graph of the function.

From the table above, it can be seen that the function only changes its value when a new integer is reached. It can be noted that the greatest integer function is a step function. Its graph is presented as follows.

Dominika is going to a movie at a local theater on her last day of vacation.

The cost to park in the theater lot is for less than an hour. An additional is charged for each hour of parking.

Next, since the greatest integer function is multiplied by each value of is multiplied by vertically stretching the spaces between each horizontal segment.

Then, since is added to the product, each segment is translated vertically units up.

Finally, the scope of the coordinate plane will be adjusted so that more steps of the graph can be seen.

During his summer vacation, Ignacio went to private math lessons. After learning about the greatest integer function, Ignacio was asked by his math tutor to graph the numbers greater than or equal to

Graph the greatest integer function. Then, determine the solution set by shading the appropriate region.

The first step to graph an inequality is to graph the border function. The border function of the given inequality is given by the greatest integer function. This function is a step function whose output is the greatest integer less than or equal to the input Note that the inequality is non-strict, so the horizontal lines are drawn as solid lines.

Finally, the following is the complete graph of the inequality that the teacher asked for.

At the beginning of this lesson, it was asked that a function for the number of soda cans in Mark’s refrigerator be written. Mark starts the week with cans in the fridge and then does the following.