SWBAT… define and evaluate functions Agenda 1. Warm-Up (10 min) 2. Review hw#3 & hw#4 (25 min)...

SWBAT… define and evaluate functionsAgenda 1. Warm-Up (10 min)2. Review hw#3 & hw#4 (25 min)

Warm-Up:

1.) How would the graph of y = |x| + 5 transform from the parent function graph, y = |x|?

The graph of y = |x| + 5 would shift 5 units up from the parent function, y = |x|.

2.) How would the graph of y = |x + 1| transform from the parent function graph, y = |x|? (Hint: It does NOT shift up 1 unit, see HW#3, problem 2)

The graph of y = |x + 1| would shift one unit to the left from the parent function, y = |x|.

Review PPT4: Functions on the Infinity website

Mon, 11/7

3. Q: How would the graph of y = -|x + 1| transform from the parent function graph, y = |x|? A: The graph of y = -|x + 1| would shift 1 unit to the left and rotate around the x-axis from the parent function, y = |x|.

(HW#3, Problem #5a)

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = -|x + 1|

The vertex, or maximum point, is (-1, 0).

Graph y = -|x + 1| by completing a table of values:

x y-2 -1 0 1 2

y =-|-2 +1| = -1 y =-|-1+ 1| = 0 y =-|0 + 1| = -1 y =-|1 +1| = -2 y =-|2 +1| = -3

y = -|x + 1| is shifted 1 unit to the left and rotated around the x-axis from the parent function, y = |x|

HW#3, Problem #6

5. Q: How would the graph of y = -|x + 1| + 3 transform from the parent function, y = |x|?

A: The graph of y = -|x + 1| + 3 would shift 1 unit to the left shift, 3 units up, and rotate around the x-axis from the parent function graph, y = |x|.

When graphing

1. Since the ordered pairs are all positive, the origin can be at the bottom left of the graph

2. Include a title on the graph

3. Include an x-axis and y-axis label

4. The x-axis has to increase by the SAME amount for each box

SWBAT… define and evaluate functions

Agenda 1. Warm-Up (5 min)2. Notes on functions (25 min)3. OYO problems (15 min)

Warm-Up:1.) Turn in HW#3 and HW#4 in the blue folder2.) Set up Cornell notes. Topic is “Functions”

HW#4: Function (page 1)

Tues, 11/8

ObjectivesToday:

1. To define a function

2. To learn function notation

3. To evaluate functions

Tomorrow:

1. To learn function mapping

2. To conduct the vertical line test

3. To find the domain and range of a function

4. To write a function as an ordered pair

Ms. Sophia Papaefthimiou

Infinity HS

What is a function?

A function is like a machine: it has an input and an output.

Function Notation The most common name is "f", but you can have

other names like "g" What goes into the function (input) is put inside

parentheses after the name of the function Example: f(x) (pronounced “f of x”)

The function is called "f“"x" goes in (input)

Question: 1.) A function is called “g” and “a” is the input. Write the function. 2.) What is the function name3.) What is the input

y f x • Output Value• Range• Dependent Variable

These are all equivalent names for the y.

• Input Value• Domain• Independent Variable

These are all equivalent names for the x.

Name of the function

Function Notation: The Symbolic Form

Function Notation

Function notation replaces the ___ in an equation with ___ Example: Given y = 3x + 2, write the equation in function

notation

f(x) = 3x + 2

Question: Write y = x2 in function notation.

y f(x)

Function Notation You used to say “y = 2x + 3; find the value of y when x = -1”

y = 2x + 3

y = 2(-1) + 3

y = -2 + 3

y = 1

Now you say “f(x) = 2x + 3; find f(-1)”

(pronounced as "f of x is two x plus three; find f of negative one")

1

-1

f(-1) = 2x + 3

Ex: If f(x) = 2x + 3 find:

1. f(4x)

2. f(-1) + f(-9)

3. f(t – 5)

OYO (On Your Own) ProblemsDirections:

If f(x) = 2x – 4 and g(x) = x² – 4x, find each value:

1. f(-3)

2. f(3x)

3. g(t) 4. f(q + 1) (Hint: f(q+1) = 2(q+1) – 4

5. f(2) + g(-2) 6. f(g(-2)) (Hint: Start from the inside out. Find g(-2) first)

Then find f(12)

Revisit our objectives

Today:

1. To define a function

2. To learn function notation

3. To evaluate functions

A function P is defined as follows:For x > 0, P(x) = x5 + x4 – 36x – 36For x < 0, P(x) = -x5 + x4 + 36x – 36

What is the value of P(-1)?A. -70B. -36C. 0D. 36

SWBAT… define and evaluate functions

Agenda 1. Warm-Up (10 min)2. Notes on functions (35 min)

Warm-Up:

If f(x) = -4x + 1 and g(x) = x², find:1.) f(-10) 2.) g(-7)3.) f(g(-7)) (Hint: Start from the inside out. Find g(-7) first)

Then find f(49))

HW#3: Evaluating functions (page 2 & 3)

Wed, 11/9

ObjectivesYesterday

1. To define a function

2. To learn function notation

3. To evaluate functions

Today

1. To learn function mapping

2. To conduct the vertical line test

3. To find the domain and range of a function

SWBAT… define and evaluate functions

Agenda 1. Warm-Up (10 min)2. Notes on functions (10 min)3. Work on hw

Warm-Up:

If f(x) = -3x + 5 and g(x) = x² – 6, find:1.) f(-8) 2.) g(-4)3.) f(g(-2)) (Hint: Start from the inside out. Find g(-2) first)

Then find f(16))

HW#3: Evaluating functions (page 1 – 4)

Thurs, 11/10

Function Mapping

A set of points or equation where every input has exactly one output.

In other words, the domain or x value can not be repeated

This is a function! There is only one arrow coming from each x.In other words, x can not be repeated

This is a function! There is only one arrow coming from each xThere is only one y for each x. It just so happens that it's always the same y for each x.

Function Mapping (cont’d)

This one is not a function.There are two arrows coming from the number 1.The number 1 is associated with two different range elements. In order words, x is repeated.

Function Mapping (cont’d)

Vertical Line Test

No mater where we drop a vertical line, if the vertical line only hits the graph once, it is a function. (Note: I didn’t say anything about the x-axis)

So, this graph is a function!

What is this function called? Quadratic function

(2, 4)(-2, 4)

y

x

Q: Is the graph a function? Explain. A: Yes, this graph is a function

because it passes the vertical line test; wherever you drop a vertical line, it will only hit the graph once.

Q: What is the graph called? A: A line

Q: What is the function called? A: A linear function

Write the equation of the line in slope-intercept form, y = mx + b.

Q: What is the slope, m? A: 3

Q: What is the y-intercept, b? A: -2

Vertical Line Test (cont’d)

y = 3x – 2

Draw a graph, that would NOT pass the vertical line test.

Exchange note books with the person next to you. Write on their notebooks if their graph passes the

vertical line test.Give them feedback

Revisit our objectives

Today:

1. To learn function mapping

2. To conduct the vertical line test

3. To find the domain and range of a function

WE MET OUR OBJECTIVES TODAY!!!!

Vertical Line Test (cont’d)

Intersect at two points These graphs are not functions

SWBAT… find the domain and range of functions

Agenda 1. Warm-Up (15 min)2. 6 practice problems (20 min)3. Exit slip (10 min)

Warm-Up:1. Write your HW in your planner for the week.2. Graph the function f(x) = 1/2x2 using a table of

values.3. Find the domain. Explain.4. Find the range. Explain.

HW#5: Evaluating functions (page 1 – 6)(Page 4 will be counted as a quiz grade)

Mon, 11/14

x f(x) =1/2x2

-4 f(-4) = 8

-2 f(-2) = 2

-1 f(-1) = 0.5

0 f(0) = 0

1 f(1) = 0.5

2 f(2) = 2

4 f(4) = 8

The domain is all real numbers.Explanation: There are no restrictions on the domain, the x value.

The range is y ≥ 0. Explanation: The graph will never be below 0.

Domain and Range

• Domain: The set of all x values in a function (width)

• Range: The set of all y values in a function (height)

What can go into a function is called the Domain

What comes out of a function is called the Range

Domain and Range (cont’d)

f(x) = x2 – 2 The domain is all real numbers.Explanation: There are no restrictions on the domain, the x value.

The range is y ≥ -2. Explanation: The graph will never be below -2.

(2, 4)(-2, 4)

(0, -2)

Domain and Range (cont’d)

Question: For the function f(x) = x2, if the domain is {1, 2, 3}, what is the range?

Domain and Range (cont’d)

Example: {(2, 4), (4, 5), (7, 3)} Question: What is the domain? What is the range? Is this a function? Explain.

Example: {(2, 4), (2, 5), (7, 3)} Questions: What is the domain? What is the range? Is this a function? Explain.

Conclusion A function relates inputs to outputs The domain is the set of all x values in a function The range is the set of all y values in a function You can use a function and its given domain, the x-value, to

find its range the y-value or f(x) value Parenthesis in function notation do not indicate

multiplication. “f(x)” means “plug in a value for x”

An input, x, and its matching output, y or f(x), together are called an ordered pair

Functions can be represented with a: 1. Table 2. Mapping 3. Equation 4. Graph

Question 1/6

Question 2/6

Suppose g(v) = v + 10 . What is g(x – 1)?

Answer: g(x – 1) = x – 1 + 10

g(x – 1) = x + 9

Question 3/6

What is the function name, input, and output of g(7)?

Answer: g is the function name7 is the inputg(7) is the output

Question 4/6

Suppose g(x) = 3x + 2. Describe, in words, what the function g does.

Answer: The function g takes an input x, multiplies by 3, and then adds 2.

Question 5/6

Write in function notation “the function g takes an input y adds 3, and then multiplies by 2.” (Hint: PEMDAS!!!!)

Answer: g(y) = 2(y + 3)

What is the domain of

(Hint: what numbers can you plug in for x to get an answer that makes sense)

1. All real numbers2. All real numbers except for 03. All real numbers except for 34. All real numbers except for -3

31

x

y

Question 6/6

Exit slip: Do the problems on the back of last week’s agenda (collected in 10 min)

HOMEWORK

HW5: Functions – Page 1 - 61. All 4 pages

2. Page 4 will count as a quiz grade

Find the domain of each function:

5

1.2

xy

174.3 xy

xy .4

xy

1.1

9.5 xy

Answers:

1. All real numbers except for 02. All real numbers except for -53. All real numbers4. All real numbers greater than or equal to -9

Question 1/10

623

125

Is this a function?

Write the below problem in your notebook

Question 2/10

If f(x) = -x2 find

a. f(3)

b. f(-3)

Question 4/10

Given f(x) = 3x – 5 and the domain is {0, 2, -1}

find the range

Question 3/10

Function f is defined by f(x) = -2x2 + 6x – 3

Find f(-2) and write as an ordered pair