DAY 5

Graphing Radical Functions Find the inverse graph of each parabola. Does the inverse exist? If not, what do you need to do in order for the graph to have an inverse? 1) 2) Restricted Domain: ________________ Restricted Domain: ________________ Range: __________________________ Range: __________________________ Inverse Function Domain: ________________________ Domain: ________________________ Range: __________________________ Range: __________________________ Finding Domain and Range of Inverse Functions

Find the inverse of each quadratic equation. Make sure to specify how you are going to restrict the domain in order for the function to have an inverse. Check your work by graphing the original quadratic function on the calculator and then the inverse of the restricted function. Principle Root. 5) 6) Restricted Domain: ________________ Restricted Domain: ________________ Range: __________________________ Range: __________________________ Inverse Function Domain: ________________________ Domain: ________________________ Range: __________________________ Range: __________________________

7) 8)

Restricted Domain: ________________ Restricted Domain: ________________ Range: __________________________ Range: __________________________ Inverse Function Domain: ________________________ Domain: ________________________ Range: __________________________ Range: __________________________

Graphing Radicals: y = x

x y

Find the Vertex Practice:

1. y = x − 4 + 8 Vertex: ( , ) 4. y = − x Vertex: ( , )

2. y = − x −1 Vertex: ( , ) 5. y = x − 7 Vertex: ( , )

3. y = 1

3x +10 Vertex: ( , ) 6. y = 3 x +1 Vertex: ( , )

1) y = − x Initial Point: ( , ) Chart

2) y = −x Initial Point: ( , ) Chart

3) y = x − 3 Initial Point: ( , ) Chart

4) y = − x + 2 + 8 Initial Point: ( , ) Chart

Graphing with an “a- value”

1) y = 2 x Initial Point: ( , ) Chart

2) y = 1

3x

Initial Point: ( , ) Chart

PRACTICE!

1) y = −2 x − 3 Initial Point: ( , ) Chart

2) y = 3 −x Initial Point: ( , ) Chart

3) y = 1

4x − 7

Initial Point: ( , ) Chart

4) y = − 1

2x + 2 + 8

Initial Point: ( , ) Chart

General Form of Radical Transformations

Given the quadratic function y = a x − h + k

If a > 0, does the graph open up or down? _________ If a < 0, does the graph open up or down? _________

If |a| > 1, does the graph have a vertical stretch or vertical shrink?________ If 0 < |a| < 1, does the graph have a vertical stretch or vertical shrink?________ What does the parameter k control? _________________________

What does the parameter h control? _________________________ What is the vertex? _________ ** “a” is NOT slope in other functions! **

QUESTION ANSWER A ANSWER B

1 What is the vertex of y = x − 2 + 5 (–2, 5) (2, 5)

2 What is the vertex of y = x −10 (1, 10) (10, 0)

3 What is the vertex of y = −4 x −1 (-1, 0) (0, -1)

4 Is the graph stretched? y = −2 x − 8 Yes No

5 What values make the graph stretched? |a| is greater than 1 a is between -1 and 1

6 What values make the graph compressed? |a| is less than 0 |a| is less than 1

7 What is the chart for: y = (x – 3)2 – 5 1 1 2 4 3 9

4 16

1 1 4 2 9 3 16 4

8 What is the chart for: y = 2 x − 3

2 1 8 2 18 3 32 4

1 2 4 4 9 6 16 8

9 Which graph shifts down 5? y = x − 5 y = x − 5

10 Which graph shifts right 5? y = x − 5 y = x − 5

11 Which graph shifts left? y = x + 2 y = x − 2

12 Reflection: y = − x Over the x-axis Over the y-axis

13 Reflection: y = −x Over the x-axis Over the y-axis

Cubic Parent Function Name: _______________________________________________ Period: ________ Graph the following cubic functions with a table. 1. f (x )= x 3

x y

-3

-2

-1

0

1

2

3

2. f (x )= −x 3

x y

-3

-2

-1

0

1

2

3

Conclusions:

1. Graph: Describe: Inflection Point________ Domain________ Range________

Sketch:

2. Graph: Describe: Inflection Point________ Domain________ Range________ Transformation:

Sketch:

3. Graph:

Describe: Inflection Point________ Domain________ Range________ Transformation:

Sketch:

4. Graph: f (x) = x − 43 +1 Describe: Inflection Point________ Domain________ Range________ Transformation:

Sketch:

Summary:

Graphing the Radical Functions

1. f (x) = x 2. f (x) = x3 3. f (x) = x + 3 − 4

4. g(x) = 3 x −1+ 3 5. r(x) = − x3 + 4 6. p(x) = 1

2−x

Graph the Parent Function Constant Function Linear Function Absolute Value Function

f (x) = c f (x) = x f (x) = x

Square Root Function Quadratic Function Cubic Function

f (x) = x f (x) = x2 f (x) = x3 Functions have the same transformations as the absolute value function y = a|x – h| + k.

Given y = a • f (x − h )+ k

if : Vertically Stretch the graph by a factor of

if : Vertically Shrink the graph by a factor of

if : Reflect the graph about the x-axis

(h, k): Translate the graph horizontally h units and vertically k units.

Let c be a positive real number. Let y = f (x) .

Vertical shift c units upward: h(x) = f (x) + c Horizontal shift right c units: h(x) = f (x − c)

Vertical shift c units downward: h(x) = f (x) − c Horizontal shift left c units:

h(x) = f (x + c)

Graph the following:

1. f (x) = x + 2 2. g(x) = x2 − 3 3. h(x) = x −1+ 4

4. y = |x – 5| + 2 5. y = (x + 3)2 + 4 6. y = x + 83 − 6 Use your calculator to verify all your graphs!