Bell Assignment

1. Graph the equation y = x3 + 3x2 – 1 on your GUT. Then use the graph to describe the increasing or decreasing behavior of the function.

2. Graph the following equation. f(x) = -½x – 6 ; x ≤ -4 x + 5 ; x > -4

1.4 Shifting, Reflecting, and Stretching Graphs

Ways to write functions:

In Algebra 2:

y = x2 + 3 y = (x+3)2

y = 3x2

In Pre-Cal

h(x) = f(x) + 3 g(x) = f(x + 3)q(x) = 3 f(x)

y = x2 Original Function f(x) = x2

Ways to write functions:

In Algebra 2:

y = x – 2 y = x -2 y = ½ x

In Pre – Cal

h(x) = f(x) – 2 p(x) = f(x – 2) q(x) = ½ f(x)

y = x f(x)= x

Shifting and Reflecting Graphs

Label Points: Notice (x, y) (-x, y) Over _____ (x, y) (x, -y) Over _____

y axis

x axis

so f(-x) reflects over the y axis because you negate the x value.

-f(x) reflects over the x axis because you negate the y value.

f(-x) means to negate the x value and therefore reflects over the y axis. -f(x) means to negate the y value and therefore reflects over the x axis.

Compare the graph of each function with the graph of f(x) = x3

In Words: In Terms of f(x) g(x) = x3 – 1 Moves down 1 Unit g(x) = f(x) – 1

p(x) = 4x3 Narrower p(x) = 4f(x)

h(x) = (x -1)3 Moves to the right 1 unit h(x) = f(x – 1)

k(x) = (x + 2)3 + 1 Moves left 2 units and 1 unit up k(x) = f(x + 2) + 1

Find an equation for each shift of f(x) = x2

g(x) = f(x) + 2

g(x) = x2 + 2

h(x) = f(x+3)

h(x) = (x + 3)2

p(x) = f(x – 4) + 2

p(x) = (x – 4)2 + 2

Reflections: h(x) = -f(x) means reflect over the x axis… why? h(x) = f(-x) means reflect over the y axis…why?

f(x) = x4

Original Graph

In terms of f(x)

In terms of x

In terms of f(x)

In terms of x

g(x) = -f(x) + 3

g(x) = -x4 + 3

h(x) = -f(x – 4)

h(x) = - (x – 4)4

Order is Important when Graphing!!!

RxSRy

(reflect over x-axis, shift, reflect over y-axis)

Graph. y = √(2-x) + 3

y = √(-x+2) + 3

Then Graph

Rewrite.

Graph. y = -1(x – 3) 3 – 4

Graph. y = -(x+2)3 + 4

Graph. y = (2-x)3 + 4

Consider the following graph.

(a) y = f(x) -1 (b) y = f(x+1) (c) y = f(x-1) (d) y = -f(x-2) (e) y = f(-x) (f) y = ½f(x) (g) y = f(2x)

Exit Pass

Describe the sequence of events. The original graph is f(x) = x3

1. g(x) = -f(x+3) – 2 2. h(x) = f(4-x) 3. p(x) = f(x-1)-3