Lesson 2 You Are the Imagineer Develop Understanding

Ready

1.

The black curve in the graph shows the graph of $f (x) = \sin (x)$ .

a.

Write the equation of the green dotted line labeled $g (x)$ .

b.

Write the equation of the purple dotted line labeled $h (x)$ .

c.

List everything you notice about the graphs of $f (x)$ , $g (x)$ , and $h (x)$ .

2.

The black curve in the graph shows the graph of $f (x) = \sin (x)$ .

a.

Write the equation of the purple line labeled $m (x)$ .

b.

Sketch in the graph of $f (x) \cdot m (x)$ .

c.

What is the equation of $f (x) \cdot m (x)$ ?

d.

Would the line $y = - 3$ also be a boundary line for your sketch? Explain.

3.

The black curve in the graph shows the graph of $f (x) = \sin (x)$ .

a.

Write the equation of the purple line labeled $b (x)$ .

b.

Sketch in the graph of $f (x) \cdot b (x)$ .

c.

What is the equation of $f (x) \cdot b (x)$ ?

d.

How is the graph of $f (x) \cdot b (x)$ different from the graph of $f (x) \cdot m (x)$ ?

e.

Would the line $y = 3$ also be a boundary line for your sketch? Explain.

Set

4.

$f (x) = x$

$g (x) = \sin (x)$

$h (x) = f (x) + g (x)$

a.

Fill in the values for $h (x)$ in the table.

$x$	$f (x)$	$g (x)$
$- 2 π$	$- 6.28$	$0$
$- \frac{3 π}{2}$	$- 4.71$	$1$
$- π$	$- 3.14$	$0$
$- \frac{π}{2}$	$- 1.57$	$- 1$
$0$	$0$	$0$
$\frac{π}{2}$	$1.57$	$1$
$π$	$3.14$	$0$
$\frac{3 π}{2}$	$4.71$	$- 1$
$2 π$	$6.28$	$0$

b.

With a smooth curve, graph

$h (x) = x + \sin (x)$

5.

$f (x) = x$

$g (x) = \sin (x)$

$k (x) = f (x) \cdot g (x)$ or $k (x) = x \cdot \sin (x)$

a.

Fill in the values for $k (x)$ in the table.

$x$	$f (x)$	$g (x)$
$- 2 π$	$- 6.28$	$0$
$- \frac{3 π}{2}$	$- 4.71$	$1$
$- π$	$- 3.14$	$0$
$- \frac{π}{2}$	$- 1.57$	$- 1$
$0$	$0$	$0$
$\frac{π}{2}$	$1.57$	$1$
$π$	$3.14$	$0$
$\frac{3 π}{2}$	$4.71$	$- 1$
$2 π$	$6.28$	$0$

b.

Now graph $k (x) = f (x) \cdot g (x)$

$k (x) = x \cdot \sin (x)$ .

Match each equation below with the appropriate graph. Describe the features of the graph that helped you match the equations.

A.	B.
C.	D.
E.	F.

6.

Equation	Graph	Recognizable Features
$f (x) = \| x^{2} - 4 \|$
$g (x) = - x + 5 \sin (x)$
$h (x) = 4 \| \sin x \|$
$d (x) = 10 - x^{2} + 5 \sin (x)$
$w (x) = - x \cdot 2 \sin (x)$
$r (x) = (2 x - 4) + \| x \|$

Go

7.

The chart names five families of functions and the parent function. The parent is the equation in its simplest form. In the right-hand column is a list of key features of the functions in random order. Match each key feature with the correct function. A key feature may relate to more than one function.

Family	Parent(s)	Key features
Linear	$y = x$	a) The ends of the graph have the same behavior. b) Each graph has a horizontal asymptote and a vertical asymptote. c) The graph only has a horizontal asymptote. d) These functions either have both a local maximum and minimum or neither a local maximum nor minimum. e) The graph is usually defined in terms of its slope and $y$ -intercept. f) The graph has either a maximum or a minimum but not both. g) As $x$ approaches $- \infty$ , the function values approach the $x$ -axis. h) The ends of the graph have opposite behavior. i) The rate of change of this graph is constant. j) The rate of change of this graph is constantly changing. k) This graph has a linear rate of change. l) These functions are of degree $3$ . m) The variable is an exponent. n) These functions contain fractions with a polynomial in both the numerator and denominator. p) The constant will always be the $y$ -intercept.
Quadratic	$y = x^{2}$
Cubic	$y = x^{3}$
Exponential	$y = 2^{x}$ , $y = 3^{x}$ , etc.
Rational	$y = \frac{1}{x}$