Lesson 7 Form Follows Function Practice Understanding

Learning Focus

Use different forms of linear and exponential functions to efficiently write equations.

Use the information given in different forms of equations to graph functions.

How do I use forms of equations for graphing linear and exponential functions?

What is the purpose of having different forms of equations?

How do I choose which form is most efficient?

Open Up the Math: Launch, Explore, Discuss

In our work so far, we have seen linear and exponential equations in many forms. Some of the forms of equations and their names are:

Linear Functions:

Equation	Name
$y = \frac{1}{2} x + 1$	Slope-intercept Form $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept
$y = \frac{1}{2} (x - 4) + 3$	Point-slope Form $y = m (x - x_{1}) + y_{1}$ , where $m$ is the slope and $(x_{1}, y_{1})$ the coordinates of a point on the line
$f (0) = 1, f (n) = f (n - 1) + \frac{1}{2}$	Recursive Formula $f (n) = f (n - 1) + D$ , Given an initial value $f (a)$ $D$ = constant difference in consecutive terms (used only for discrete functions)

Equation

Name

$y = \frac{1}{2} x + 1$

Slope-intercept Form

$y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept

$y = \frac{1}{2} (x - 4) + 3$

Point-slope Form

$y = m (x - x_{1}) + y_{1}$ , where $m$ is the slope and $(x_{1}, y_{1})$ the coordinates of a point on the line

$f (0) = 1, f (n) = f (n - 1) + \frac{1}{2}$

Recursive Formula

$f (n) = f (n - 1) + D$ ,

Given an initial value $f (a)$

$D$ = constant difference in consecutive terms (used only for discrete functions)

Exponential Functions:

Equation	Name
$y = 10 {(3)}^{x}$	Explicit Form $y = a {(b)}^{x}$
$f (0) = 10, f (n + 1) = 3 f (n)$	Recursive Formula $f (n + 1) = R f (n - 1)$ Given an initial value $f (a)$ $R$ = constant ratio between consecutive terms (used only for discrete functions)

Equation

Name

$y = 10 {(3)}^{x}$

Explicit Form

$y = a {(b)}^{x}$

$f (0) = 10, f (n + 1) = 3 f (n)$

Recursive Formula

$f (n + 1) = R f (n - 1)$

Given an initial value $f (a)$

$R$ = constant ratio between consecutive terms (used only for discrete functions)

Knowing a number of different forms for writing and graphing equations is like having a mathematical toolbox. You can select the tool you need for the job, or in this case, the form of the equation that makes the job easier. Any master builder will tell you that the more tools you have the better. In this task, we’ll work with our mathematical tools to be sure that we know how to use them all efficiently. As you model the situations in the following problems, think about the important information in the problem and the conclusions that can be drawn from it. Is the function linear or exponential? Does the problem give you the slope, a point, a ratio, a $y$ -intercept? Is the function discrete or continuous? This information helps you to identify the best tools and get to work!

C.

neither

d.

The domain of the function is:

3.

a.

Write the equation of the geometric sequence with a constant ratio of $5$ and a first term of $- 3$ . Name the form of the equation you wrote and why you chose to use that form.

b.

This function is:

A.

linear

Name the form of the equation you wrote and why you chose to use that form.

b.

This function is:

A.

linear

B.

exponential

C.

neither

c.

This function is:

A.

continuous

B.

discrete

C.

neither

d.

The domain of the function is:

6.

Yessica’s science fair project involved growing some seeds to see what fertilizer made the seeds grow fastest. One idea she had was to use an energy drink to fertilize the plant. (She thought that if energy drinks make people perky, energy drinks might have the same effect on plants.) This is the data that shows the growth of the seed each week of the project.

Week	$1$	$2$	$3$	$4$	$5$
Height (cm)	$1.7$	$2.9$	$4.1$	$5.3$	$6.5$

a.

Write the equation that models the growth of the plant over time.

Name the form of the equation you wrote and why you chose to use that form.

b.

This function is:

A.

linear

B.

exponential

C.

neither

c.

This function is:

A.

continuous

B.

discrete

C.

neither

d.

The domain of the function is:

An equation gives us information that we can use to graph the function. Pick out the important information given in each of the following equations and use the information to graph the function.

7.

$y = \frac{1}{2} x - 5$

What do you know from the equation that helps you to graph the function?

8.

$y = 2^{n}$

What do you know from the equation that helps you to graph the function?

9.

$y = - 2 (x + 6) + 8$

What do you know from the equation that helps you to graph the function?

10.

$f (1) = - 5, f (n) = f (n - 1) + 1$

What do you know from the equation that helps you to graph the function?

Ready for More?

Here’s a challenge: Choose one of the scenarios below to write your own problem.

Write a linear context that is continuous and would make slope-intercept the most efficient form for modeling the context.
Write a linear table for a discrete function that would make a recursive equation the most efficient form for modeling the context.
Write a context for a continuous exponential function that would make an explicit equation the most efficient form for modeling the context.

Takeaways

Linear Functions:

Name	What do you need to use it to write equations?	Equation
Slope-intercept Form $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept		$y = \frac{1}{2} x + 1$
Point-slope Form $y = m (x - x_{1}) + y_{1}$ , where $m$ is the slope and $(x_{1}, y_{1})$ the coordinates of a point on the line		$y = \frac{1}{2} (x - 4) + 3$
Recursive Formula $f (n) = f (n - 1) + D$ , Given an initial value $f (a)$ $D$ = constant difference in consecutive terms (used only for discrete functions)		$f (0) = 1$ , $f (n) = f (n - 1) + \frac{1}{2}$

Name

What do you need to use it to write equations?

Equation

Slope-intercept Form

$y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept

$y = \frac{1}{2} x + 1$

Point-slope Form

$y = m (x - x_{1}) + y_{1}$ , where $m$ is the slope and $(x_{1}, y_{1})$ the coordinates of a point on the line

$y = \frac{1}{2} (x - 4) + 3$

Recursive Formula

$f (n) = f (n - 1) + D$ ,

Given an initial value $f (a)$

$D$ = constant difference in consecutive terms

(used only for discrete functions)

$f (0) = 1$ ,

$f (n) = f (n - 1) + \frac{1}{2}$

Exponential Functions:

Name	What do you need to use it to write equations?	Equation
Explicit Form $y = a {(b)}^{x}$		$y = 10 {(3)}^{x}$
Recursive Formula $f (n) = R f (n - 1)$ Given an initial value $f (a)$ $R$ = constant ratio between consecutive terms (used only for discrete functions)		$f (0) = 10$ , $f (n) = 3 f (n - 1)$

Name

What do you need to use it to write equations?

Equation

Explicit Form

$y = a {(b)}^{x}$

$y = 10 {(3)}^{x}$

Recursive Formula

$f (n) = R f (n - 1)$

Given an initial value $f (a)$

$R$ = constant ratio between consecutive terms

(used only for discrete functions)

$f (0) = 10$ ,

$f (n) = 3 f (n - 1)$

Lesson Summary

In this lesson, we summarized our work with writing equations for linear and exponential functions. We worked on strategically selecting a useful form for the context by identifying the information about the type of change and the initial values, wherever they are.

Retrieval

1.

Compare the functions $h (x) = 5 {(2)}^{x}$ and $g (x) = 5 + 2 x$ . What is similar and what is different about them?

a.

What is similar and different about the graph for each?

b.

How do their $y$ -intercepts compare?

c.

What kind of functions are $h (x)$ and $g (x)$ ?

d.

What type of growth does each function have?

Solve each equation. Show your work and justify your steps.

2.

$x + 6 = 32$

3.

$- 4 x = 52$

4.

$7 x = - 35$