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Unit 3 Name______________________________

Functions & their Graphs

|Day 1 |Recognizing & Evaluating Functions |

|Day 2 |Graphing Quadratic & Absolute Value Functions |

|Day 3 |Graphing Exponential, Square Root, & Cube Root Functions |

|Day 4 |Graphing all Types |

|Day 5 |Quiz Review |

|Day 6 |QUIZ |

|Day 7 |Domain, Increasing & Decreasing |

|Day 8 |Graphing Functions from Set Builder Notation & Graphing over a specific Domain |

|Day 9 |Practice |

|Day 10 |Unit 3 Review |

|Day 11 &12 |Unit 3 TEST |

|Day 13 |Regents Questions |

[pic]

Unit 3 Vocabulary:

|Word |Meaning |Where to find more info |

|[pic] | | |

|[pic] | | |

|[pic] or [pic] | | |

|Axis of Symmetry | | |

|(AOS) | | |

|Decreasing | | |

|Domain | | |

|Function | | |

|Increasing | | |

|[pic] | | |

|Ordered pair | | |

|Parabola | | |

|[pic] | | |

|Range | | |

|Relation | | |

|W | | |

|[pic] | | |

A relation is simply a set of ordered pairs.

The domain refers to all of the x-values.

The range refers to all of the y-values.

A function is a relation in which each member of the ____________ is paired with one and only one member of the ____________. (x-values cannot repeat!)

☼ Ordered Pairs:

Function: {(2, 3), (3, 4), (5, 1), (6, 2), (8, 7)}

Not a Function: {(1, 4), (2, 3), (5, 4), (2, 6)}

☼ Vertical Line Test:

Draw a few vertical lines on the graph.

6) 7)

Function Notation

Function Notation, [pic], is basically instead of saying something like [pic], we say [pic]. This is saying that the result of [pic] is a “function of x”. The x value represents the domain and the answer (the y – value) represents the range.

So when they ask us to determine the value of [pic] in the function: [pic], we simply plug 2 into the equation in the “x” spot.

[pic]

Let’s try some:

|9) Determine the value of [pic] in the function [pic]. |10) Determine the value of [pic] in the function [pic]. |

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|11) Find the range of the function [pic] for the domain {–1, 0, 1}. |

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Let [pic], and let [pic], and suppose a, b, c, and h are real numbers. Find the value of each function for the given input.

|12) [pic] |13) [pic] |

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|14) [pic] |15) [pic] |

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|16) [pic] |17) [pic] |

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Determine whether the following are functions or relations and explain your answer!

|a) |b) |

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|c) |d) |

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|e) |f) |

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|g) |h) |

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♣ Parabolas are graphs of quadratic (x2) equations.

Quadratic equation in standard form: [pic]

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|♣ If “a” is ______________________, the |♣ If “a” is ______________________, the |

|parabola will “___________________” and have a _________________ |parabola will “___________________” and have a _________________ |

|point. |point. |

|ex) [pic] |ex) [pic] |

|Sketch it here: |Sketch it here: |

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The axis of symmetry (AOS) is a line that cuts the parabola in half so the two sides are mirror images.

When we graph we want to know where this spot is so that we build our table around it and end up with a graph that shows us the whole parabola.

To make the table, put the AOS value in the middle of the x column, then three values smaller and three values larger.

[pic]

4) [pic]

a) Make table:

5) [pic]

[pic]

An Absolute Value graph is in the shape of a V.

An Absolute Value function contains [pic].

6) Graph: [pic]

in calculator: ____________

make table:

7) Sketch: [pic]

in calculator: ____________

An exponential function has a variable as the exponent.

1) Sketch: [pic] 2) Sketch: [pic]

in calculator: ____________ in calculator: ____________

A Square root function contains [pic]

3) Graph: [pic] in calculator: ____________

❖ Why can’t negative values be used for a square root function?

_____________________________________

A cube function contains [pic]

4) Graph: [pic] in calculator: ____________

A cube root function contains [pic]

5) Graph: [pic] in calculator: ____________

❖ Why can negative values be used for a cube root function?

_____________________________________

Domain, Increasing, Decreasing

Domain: _____________________________________________________________________

Range: _______________________________________________________________________

|[pic] |[pic] |[pic] |[pic] |[pic] or : |

Domain can be written in 3 ways: Intervals of Increasing/Decreasing:

|Inequality |[pic] |[pic] |*Increasing/Decreasing intervals only use parentheses* |

|Interval Notation |[pic] | |Increasing Interval: |

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|Set Builder Notation |[pic] | |Decreasing Interval: |

| | | |n/a |

Examples:

|1) |2) |

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|Domain: _______________ |Domain: _______________ |

|Increasing: _______________ |Increasing: _______________ |

|Decreasing: _______________ |Decreasing: _______________ |

|3) |4) |

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|Domain: _______________ |Domain: _______________ |

|Increasing: _______________ |Increasing: _______________ |

|Decreasing: _______________ |Decreasing: _______________ |

|5) |6) |

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|Domain: _______________ |Domain: _______________ |

|Increasing: _______________ |Increasing: _______________ |

|Decreasing: _______________ |Decreasing: _______________ |

Using your graphing calculator, graph the functions. Fill in the boxes.

|Example |Domain |Increasing |Decreasing |

| | |Interval |Interval |

|7) f(x) = 3x + 1 | | | |

|8) y = 2x2 + 2x +1 | | | |

Read each situation and decide what the domain is represented in words, then write it out in math.

| |Situation |Domain |

|9) |The relationship between human years and dog years is given by|Words: |

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| |f(x) = 7x. | |

| | |Math: |

|10) |At a salon, Sue can rent a station for $10.00 per day plus |Words: |

| |$3.00 per manicure. The amount she would pay is given by f(x)| |

| |= 3x + 10. | |

| | |Math: |

|11) |Peter needs to fill up his truck with gasoline to drive to and|Words: |

| |from school next week. Gas costs $2.79 per gallon, and his | |

| |truck holds a maximum of 28 gallons. | |

| | |Math: |

|12) |Jessie is parking in a parking garage for a concert. It costs|Words: |

| |$6 for the first 2 hours, an additional $2 for each additional| |

| |hour, with a maximum of $24 per day. | |

| | |Math: |

Graphing with a Specific Domain

Let’s recall some symbols from set builder notation and learn some new ones.

|[pic] |Set of Real numbers (all numbers that are not imaginary) |

|[pic] |Set of Integers {–3, –2, –1, 0, 1, 2, 3} |

|W |Set of WhOle numbers {0, 1, 2, 3, …} |

|[pic] |Set of Natural numbers, or Counting numbers {1, 2, 3, …} |

|[pic] |Is an element of |

|[pic] |Is not an element of |

|[pic] |Such that |

Using our knowledge of set builder notation,

we will begin to read functions using this notation.

For example:

We are used to writing linear equations as y = 3x + 1

We now know that lines written like that are functions and CAN be written as f (x) = 3x + 1

We know that y is the same as f(x)

So, [pic] is the same as [pic]

SO, we can also write this same function in set builder notation which would look like:

This could also say: [pic]

Graph these functions given in Set Builder Notation.

|1) [pic] |2) [pic] |

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|[pic] |[pic] |

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|3) [pic] |4) [pic] |

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|[pic] |[pic] |

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|5) [pic] |6) [pic] |

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| |[pic] |

|[pic] | |

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|7) [pic] |8) [pic] |

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|[pic] |[pic] |

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YOU CANNOT HAVE AN X VALUE MATCHED

UP WITH MORE

THAN ONE

Y VALUE!!!

YES

NO

It IS a function!

It IS NOT a function! (it’s a relation)

Does each line pass through the graph only once?

Don’t forget about Aunt Sally!!

[pic]

2

4

6

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|X |0 |1 |2 |3 |

|Y |2 |2 |2 |2 |

|X |3 |5 |4 |3 |

|Y |2 |1 |9 |3 |

Axis of Symmetry

Vertex

|x |[pic] |

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|x |[pic] |

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|x |[pic] |

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|[pic] |[pic] |

|–3 | |

|–2 | |

|–1 | |

|0 | |

|1 | |

|2 | |

|[pic] |[pic] |

|–2 | |

|–1 | |

|0 | |

|1 | |

|2 | |

|3 | |

|x |[pic] |

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Function Notation

Domain Variable

Function being graphed

Domain

[pic]

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