Lesson 1 Growing Roots Develop Understanding

Ready

A direct variation function is written as $y = k x$ , where $k$ is a nonzero constant. The $k$ is called the constant of variation. If you rewrite $y = k x$ as $\frac{y}{x} = k$ , you can see that $k$ is a constant ratio or a constant rate. The constant $k$ is also called the constant of proportionality.

Identify whether the given equation represents a direct variation function. If it does, state the constant of variation $k$ .

1.

$y - 7 x = 0$

Yes/no?

$k$ ?

2.

$- 2 x + 5 y = 0$

Yes/no?

$k$ ?

3.

$5 x + y = 10$

Yes/no?

$k$ ?

4.

$17 y = - 34 x$

Yes/no?

$k$ ?

State whether or not each table represents a direct variation relationship.

If it does, write the equation in $y = k x$ form. How does $y = k x$ form compare to $y = m x + b$ form?

5.

$x$	$y$
$1$	$7$
$3$	$21$
$5$	$35$
$7$	$49$

6.

$x$	$y$
$- 5$	$- 15$
$- 2$	$0$
$10$	$30$
$15$	$45$

7.

$x$	$y$
$2.5$	$- 10$
$5$	$- 20$
$7.5$	$- 30$
$10$	$- 40$

8.

$x$	$y$
$- 2$	$- 7$
$- 1$	$- 5$
$0$	$- 3$
$1$	$- 1$

9.

a.

Select all graphs that show a direct variation function.

A.

B.

C. b.

Identify the features of a direct variation graph.

Set

10.

Let $f (x) = \sqrt{x}$ be a continuous function. Make a table of values for $f (x) = \sqrt{x}$ . Then graph the function.

$x$	$0$	$0.5$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$16$
$f (x)$

11.

The graph in problem 10 represents the square root function. Select all of the features that apply to the graph of the square root function. If you choose a feature that applies to the square root function, write the specific value(s) that describe the feature.

$y$ -intercept

$x$ -intercept

Maximum

Minimum

Domain

Range

Contiuous

Discrete

Interval of increase

Interval of decrease

Constant rate of change

Rate of change is constantly changing

12.

Explain why the domain of $f (x) = \sqrt{x}$ does not include negative numbers.

13.

Four secant lines have been drawn on the graph of $f (x) = \sqrt{x}$ . Find the average rate of change of $f (x) = \sqrt{x}$ between $(0, 0)$ and the following points: $(1, 1)$ , $(4, 2)$ , $(9, 3)$ , and $(16, 4)$ . What do the different rates of change of the secant lines tell you about the rate of change of the square root function?

Go

Describe the transformation on each parabola. Then graph the function.

14.

$f (x) = {(x - 5)}^{2} + 1$

Description:

15.

$f (x) = \frac{1}{2} {(x)}^{2} - 3$

Description:

16.

$f (x) = - {(x + 2)}^{2} + 6$

Description:

17.

$f (x) = 3 {(x - 1)}^{2} + 2$

Description: