algebra unit 14 all functions the graphs of mixed...

ALGEBRA UNIT 14 ALL FUNCTIONS

THE GRAPHS OF MIXED FUNCTIONS (DAY 1)

In this unit, we will remember how to graph some old functions and discover how to

graph some new ones.

EX1: Graph and label the equation f(x) = x2 on the set of axis below

Key Features:

Function Name:___________

Domain:

Range:

Are there any Key Points to recognize in the graph?

On the same set of axes above, graph and label the function 3)2x()x(h 2

by creating a table of values

What happened from the f(x) graph to the h(x) graph

How did the equation change in h(x)?

Did the domain and range change for the h(x) function? Explain

Which function f(x) or h(x) would be considered the parent function? Explain

x f(x)

x h(x)

2

EX2: Graph and label the equation g(x) = x on the set of axis below. Create a table of

values.

Key Features:

Function Name______________

Domain:

Range:


On the same set of axes above, graph and label the function 35x)x(w

by creating a table of values

What happened from the g(x) graph to the w(x) graph

How did the equation change in w(x)?

Did the domain and range change for the w(x) function? Explain

Which function g(x) or w(x) would be considered the parent function? Explain

x g(x)

x w(x)

3

EX3: Graph and label the equation f(x) = x3 on the set of axis by creating a table of values.

Key Features:


Domain:

Range:


EX4: Graph and label the equation g(x) = 3 x on the set of axis by creating a table of values.

Key Features:


Domain:

Range:


x f(x)

x f(x)

4

THE TRANSLATIONS OF MIXED FUNCTIONS (DAY 2)

Name the type of function graphed below. Identify the parent function equation for each graph.

EX1: Graph and label the equation x)x(f on the set of axis by creating a table of values.

Key Features:


Domain:

Range:


x f(x)

xy VERSUS 2xy

Similarities Differences

5

VERTICAL AND HORIZONTAL MOVEMENTS

Given the following functions, identify its parent function. Graph both in your calculator to

detect key features of the graph

EX 2: Given: g(x) = 3x Parent Function:__________________

Domain: Key Point (vertex):

Range: What happened from the parent function to g(x)?

EX3: Given: h(x) = (x + 3)2 + 2 Parent Function:__________________


Range: What happened from the parent function to h(x)?

EX4: Given: 41x)x(f 3 Parent Function:__________________


Range: What happened from the parent function to f(x)?

Are there any correlations between the movement of the graph and its key point in

the graph?

Are there any correlations between the equation and its vertex? What do you notice?

A horizontal movement affects which part (x or y) in the equation? What do you

notice?

A vertical movement affects which part (x or y) in the equation? What do you notice?

Now Without using your calculator: Determine the vertex and describe the movement that

was applied to its parent function to get of the function 35x)x(r

6

WRITING EQUATIONS OF FUNCTIONS WITH TRANSLATIONS

HORIZONTAL AND VERTICAL SHIFTS (DAY 3)

Without using your calculator: Determine the vertex and describe the movement that was

applied to its parent function to get of the function 24x)x(w

Vertex:_____________ Movement:______________________

What happened to the x value from the equation to the vertex point?

What happened to the y value from the equation to the vertex point?

EX1: Given the function g(x) = x3 + 2. What is the parent function f(x)? __________________

o What happened to your graph from its parent graph f(x)?

o What is the new “KEY POINT” of g(x)?________

o Write the translation rule applied to the parent function f(x) to get g(x) using

FUNCTION NOTATION? ____________________________

EX2: Given the function h(x) = (x – 5)2 + 2 What is the parent function m(x)? _______________

o What happened to your graph from its parent graph m(x)?

o What is the new “KEY POINT” of h(x)?________

o Write the translation rule applied to the parent function m(x) to get h(x) using

FUNCTION NOTATION? ____________________________

Writing Movement Symbolically using Translation Rule:

Writing Movement Symbolically using FUNCTION NOTATION:

7

EX3: Given the function x3)x(f

If 2)x(f)x(g , how is the graph of f(x) translated to form the graph g(x)?

If )4x(f)x(h , how is the graph of f(x) translated to form the graph of h(x)

EX4: The vertex of the parabola represented by 3x4x)x(f 2 has the coordinates (2, -1).

Find the coordinates of the vertex of the parabola defined by ).2x(f)x(g Explain

how you arrived at your answer.

EX5: Write an equation d(x) for the image of the graph 3 x)x(f if f(x) is shifted 3 units left

and 4 units up.

EX6: Write an equation k(x) for the image of the graph 2x)x(w if w(x) is translated 2 units

right and 7 units down.

EX7: If the graph y = 25x is translated left 3 and down 9, what is the new equation?

EX8: If the graph y = 3 1+x – 4 is translated right 6 and up 7, what is the new equation?

EX9: Which equation represents the function shown in the

accompanying graph?

(1) 1x)x(f (3) 1x)x(f

(2) 1x)x(f (4) 1x)x(f

8

STRETCHING & SHRINKING FUNCTIONS (DAY 4) Complete the table below and graph each function on the axes below

X f(x) = x (parent function)

g(x) = 4 x h(x) = 2

1x k(x) = -4 x p(x) = -

2

1x

-2

-1

0

1

2

What happened in the graph x4)x(g ?

What happened in the graph x2

1)x(h ?

What happened in the graph x4)x(k

LET’S SUMMARIZE: WHEN GIVEN THE FUNCTION

When a > 1 (________ __________), the graph is______________________________________________

When 0 < a < 1 (__________ __________), the graph is________________________________________

When -1 < a< 0, (__________ __________), the graph is_______________________________________

When a < -1, (__________ __________), the graph is___________________________________________

9

EX1. Tiny graphed the function x)x(f and x3)x(g in her calculator. She said that the

g(x) function got wider. Is her statement correct? Explain your reasoning.

EX2. Given the function 1)4x(2)x(g 2 and 2x)x(h , describe what happened to h(x) to

become g(x).

EX3. The graph of a quadratic function 2x)x(d has been translated 3 units to the right,

vertically stretched by a factor of 4, and moved 2 units up. Write the formula for the

function a(x) that defines the transformed graph.

EX4. The graph of an absolute value function x)x(f has been translated 6 units to the left,

vertically shrunk by a factor of 2

1, reflected over the x-axis, and moved down 5 units.

Write the formula for the function g(x) that defines the transformed graph.

EX5. How does the graph of 1)2x(3)x(f 2 compare to the graph of 2x)x(g ?

(1) The graph of f(x) is wider than the graph of g(x) and its vertex is moved to the left 2

units and up 1 unit

(2) The graph of f(x) is narrower than the graph of g(x), and its vertex is moved to the

right 2 units and up 1 unit

(3) The graph of f(x) is narrower than the graph of g(x), and its vertex is moved to the

left 2 units and up 1 unit

(4) The graph of f(x) is wider than the graph of g(x) and its vertex is moved to the right

2 units and up 1 unit

10

FORMS OF THE QUADRATIC EQUATIONS

VERTEX FORM VS STANDARD FORM (DAY 5)

Standard Form of a Quadratic Equation: _________________________________________

Use this form to easily determine if graph is ___________ or ___________.

This form is best for _______________ equations in order to solve for the zeros.

Vertex Form of a Quadratic Equation: ____________________________________________

Use this form to quickly find the ____________ of the graph. Describe the any

____________ on the graph. Identify if the graph either ____________ or ____________

vertically.

To get a quadratic function into VERTEX FORM:

Use the procedure for ___________________ _____ ______________ to convert the form.

EX1: A quadratic function is defined by 1x12x2)x(g 2 . Rewrite this equation into vertex

form

g(x) = 2x2 + 12x + 1 Steps to Convert Quadratic Equation from

Standard Form to Vertex Form:

1. Divide by leading coefficient

2. Group x terms and move c# to other side leaving a

filler space on each side

3. Fill in the filler space by taking B# half it, square it,

add it to both sides

4. Write expression as binomial squared NEW STEPS FOR VERTEX FORM:

11

EX2: Rewrite the equation 1x4x2)x(g 2 into vertex form by completing the square and

then answer the follow-up questions.

a. What is the vertex of the function?

b. What is the axis of symmetry?

c. What is the domain of the function?

d. What is the range of the function?

e. Write a translation rule in function notation that would turn f(x) = x2 into g(x)?

EX3. Given the function 9x8x)x(h 2 , state whether the vertex represents a maximum

or minimum point for the function. Explain your answer.

Rewrite h(x) in vertex form by completing the square

EX4. Match the appropriate equations with their graphs.

a. 5x6xy 2 g(x)

b. 1)2x(3y 2 h(x)

c. x8x2y 2 k(x)

d. 3)4x(y 2 p(x)

12

DETERMINING FUNCTIONS FROM TABLES (DAY 6)

Identify the function graphed below, write the parent function, and identify the degree of

the function.

EX1. Complete the table below and determine the function that is being represented.

Justify your answer by identifying the degree of the function. Write an equation to

represent the function identified by the table.

x y

1 2

2 9

3 16

4 23

5

Degree of a function: determined by the ___________ exponent in the function. Indicates

the number of _____________ in a problem. Graphically it shows the number of times it

_____________ the x-axis, so it identifies the number of _____________ in the graph.

Degree of functions also determine the number of levels (columns) needed to find

the ______________ _____________________ . The level (column) that the common

difference appears at indicates the degree of the function.

13

EX2. Complete the table below and determine the function that is being represented.

Justify your answer by identifying the degree of the function. Write an equation to

represent the function identified by the table.

EX 3. Given table below determine the function that is being represented. Write an

equation to represent the function identified by the table.

EX4: Given table below determine the function that is being represented. Write an

equation to represent the function identified by the table.

x y

1 6

2 18

3 54

4 162

5

x y

-3 44

-2 31

-1 20

0 11

1 4

x y

-2 -5

-1 -4

0 -5

1 -2

2 11

14

CREATING QUADRATIC EQUATIONS ALGEBRAICALLY

GIVEN SPECIFIC POINTS (DAY 7)

Determine the number of quadratic equations that could be found with the following

information.

Ex 1: How many different quadratics can you draw through

the points (0, 4) & (1, 9)?


the points (0, 4), (1, 9) and (-3, 1)?


the points (0, 4), (1, 9), (-3, 1) and (2, 5)?

Conclusions:

How many points are required to be able to create a single quadratic equation?

Are any of the points special in relation to graphing functions?

15

EX1: Write a quadratic model that goes through the points (0, 4), (1, 9), and (-3, 1).

Only an algebraic solution accepted

How could you check to see if you created the correct quadratic equation?

ALGEBRAIC PROCEDURE FOR FINDING A QUADRATIC EQUATION GIVEN 3 POINTS

1. Identify the parts needed for the quadratic equation

2. Plug the point that represents the y-intercept into the generic quad equation to

solve for 1 missing part of the equation (c #)

3. Using the c# found, plug the other points given into the generic equation to create

2 new linear equations.

4. These 2 linear equations form a system the needs to be solve for the remaining parts

of the quadratic.

5. Once all missing parts are found (a, b, and c#) write the quadratic equation.

16

EX 2: An experiment was conducted in science class to see the distance from the ground

for a watermelon at specific time frames when dropped from a roof of the building.

Below is the data collected by the class.

Write an equation that models the collected data. Only an algebraic method is accepted.

Explain how you determined your equation.

Time (t) 0 1 2 3 4

Height

f(t) 300 284 236 156 44

17

PIECEWISE & STEP FUNCTIONS (DAY 8)

Piecewise Function: Graphs that are made up of “pieces” of different functions defined

over specific intervals.

EX1: Graph the following function on the axes

3x,2x2

3x0,5x)x(f

2

TO GRAPH PIECEWISE FUNCTIONS:

1. Create a table of values for each equation separately

VERY IMPORTANT: Only use x values that are SPECIFICALLY defined for each

function.

2. Plot correct points (if more are plotted than are required points are DEDUCTED)

3. Correct Endpoints for each Function (open or closed circles)

VERY IMPORTANT: Make sure you have the correct end point for each function

18

EX2: Graph the following function on the set of accompanying axes

1x3,x

8x1,4)x(g

EX3: A function is graphed below on the set of axes

to the right. Which function is related to the

graph?

(1)

1x,2x

1x,x)x(f

2

(3)

1x,

2

1x

2

1

1x,x)x(f

2

(2)

1x,

2

9x

2

3

1x,x)x(f

2

(4)

1x,7x2

1x,x)x(f

2

EX4: What is the value of f(2) when

1x,x2

1x,1xx3)x(f

2

(1) 4 (3) 11

(2) 7 (4) 13

19

Step Function: Functions that increase or decrease from one constant to another. It looks

like a staircase when graphed!

EX5: Graph the following function.

9x6,8

6x3,6

3x0,4

0x3,2

)x(f

EX 6: During a snowstorm, a meteorologist tracks the amount of accumulating snow. For

the first three hours of the storm, the snow fell at a constant rate of one inch per hour.

The storm then stopped for two hours and then started again at a constant rate of

one-half inch per hour for the next four hours.

On the grid below, draw and label a graph that

models the accumulation of snow over time using

the data the meteorologist collected.

Write the equations for the different pieces of the function graphed.

algebra unit 14 all functions the graphs of mixed...

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