Lesson 6 Another Day Solidify Understanding

Jump Start

Which One Doesn’t Belong? Examine each function and determine which one is not like the others. Be prepared to give a mathematical reason for your choice.

$x$	$f (x)$
$- 2$	$1$
$- 1$	$- 1$
$0$	$- 3$
$1$	$- 5$
$2$	$- 7$
$3$	$- 9$

$x$	$f (x)$
$- 2$	$- 3$
$- 1$	$- 1$
$0$	$1$
$1$	$3$
$2$	$5$
$3$	$7$

$g (x) = 2 x$

Reason

Learning Focus

Write and graph equations of functions.

Compare the graphs of related functions.

What is the effect on the graph when a number is added to an equation of a function?

Open Up the Math: Launch, Explore, Discuss

Every day when Rashid’s mother comes home from work, he asks her how her day was and she says, “Another day, another dollar.” Being a mathematical thinker, Rashid started imagining the function that would model his mother’s saying. His thinking went something like this:

“On day $0$ , she hasn’t worked, so she has $0$ .

On day $1$ , she works and gets $$ 1$ .

On day $2$ , she gets another dollar, so she has $$ 2$ .

One day $3$ , she gets another dollar, so she has $$ 3$ .”

Rashid visualized this graph of the function:

1.

Model the function that describes this pattern with a table and explicit equation.

As Rashid pursued this thinking, he wondered what would happen to his function if he started with $$ 5$ on day $0$ and then earned a dollar a day.

2.

Model this situation with a table, graph, and explicit equation.

3.

Compare the graph in problem 2 to Rashid’s original graph. What do you notice?

Rashid thought about all the bills that his family needed to pay and wondered what the function would be if the situation changed so that on day $0$ , they owed $$ 10$ and then added a dollar a day.

4.

Write the explicit equation and graph the function that models this situation.

5.

Compare the three graphs and equations. What do you notice?

Rashid thought, “A dollar a day is no way to get ahead. What if I start with $$ 1$ on day $0$ and double my dollars each day? That means I’d have $$ 2$ on day $1$ , $$ 4$ on day $2$ , and $$ 8$ on day $3$ . I’d come home from work and say, “Another day, another double.”

6.

Let $D (t)$ be the function that models this situation. Represent $D (t)$ with a table, graph, and equation.

7.

Rashid wondered what would happen to this function if he added $$ 3$ to every output. What’s your prediction?

8.

Try it! Use an equation, a table, and a graph to model the function that adds $3$ to every output of $D (t)$ .

9.

How does the graph of $D (t)$ compare to the new function, $D (t) + 3$ ?

10.

If $f (x) = x$ , what do you know about the graph of $f (x) + b$ ? ( $b$ represents any constant number).

11.

If $D (t) = 2^{t}$ what do you know about the graph of $D (t) + b$ ? ( $b$ represents any constant number).

12.

Compare the features of $f (x)$ and $g (x)$ .

$f (x) = {(\frac{1}{2})}^{x} + 1$

13.

Compare the features of $h (x)$ and $m (x)$ .

$x$	$m (x)$
$- 2$	$\frac{1}{9} + 1 = \frac{10}{9}$
$- 1$	$\frac{1}{3} + 1 = \frac{4}{3}$
$0$	$1 + 1 = 2$
$1$	$3 + 1 = 4$
$2$	$9 + 1 = 10$
$3$	$27 + 1 = 28$

Ready for More?

The graph of $f (x)$ is shown below. Sketch the graph of $g (x) = f (x) - 3$

Takeaways

Effects of a vertical translation on the linear parent function, $f (x) = x$ :

Effects of a vertical translation on an exponential parent function, $f (x) = 2^{x}$ :

Adding Notation, Vocabulary, and Conventions

Parent function:

Vertical translation:

Vocabulary

Lesson Summary

In this lesson, we learned that adding or subtracting a number to a function results in a vertical translation of the graph. In a vertical translation, functions do not change their shape, they are simply shifted up or down.

Retrieval

Solve the equation.

1.

$- 7 x + 30 = 72$

2.

$- 17 + 4 x = 11$

3.

Find each of the features of the function shown in the graph.

Domain:

Range:

Increasing:

Decreasing:

Max:

Min:

$x$ -int:

$y$ -int: