1 function notation & evaluating...
TRANSCRIPT
Warm up• It’s Hat Day at the Braves game and every child 10 years
old and younger gets a team Braves hat at Gate 7. The policies at the game are very strict. ▫ Every child entering Gate 7 must get a hat. ▫ Every child entering Gate 7 must wear the hat. ▫ Only children age 10 or younger can enter Gate 7. ▫ No child shall wear a different hat than the one given to
them at the gate.
1. What might be implied if all the rules were followed but there were still children 10 years old and younger in the ballpark without hats?
Coordinate Algebra
UNIT QUESTION: How can we use real-world situations to construct and compare linear and exponential models and solve problems? Standards: MCC9-12.A.REI.10, 11, F.IF.1-7, 9, F.BF.1-3, F.LE.1-3, 5 !Today’s Question: What is a function, and how is function notation used to evaluate functions? Standard: MCC9-12.F.IF.1 and 2
Functions vs Relations
Domain•x – coordinates •Independent variable •Input
Range•y – coordinates •Dependent variable •Output
Relation•Any set of input that has an output
Function•A relation where EACH input has exactly ONE output !
•Each element from the domain is paired with one and only one element from the range
Revisit the warm up:• It’s Hat Day at the Braves game and every child 10 years
old and younger gets a team Braves hat at Gate 7. The policies at the game are very strict. ▫ Every child entering Gate 7 must get a hat. ▫ Every child entering Gate 7 must wear the hat. ▫ Only children age 10 or younger can enter Gate 7. ▫ No child shall wear a different hat than the one given to
them at the gate.
1. What is the gate’s input?
!2. What is the gate’s output?
Going in: Children 10 & younger without hats
Coming out of Gate 7: Children 10 & younger WITH hats
How do I know it’s a function?
•Look at the input and output table – Each input must have exactly one output. •Look at the Graph – The Vertical Line test: NO vertical line can pass through two or more points on the graph
Function or relation?Example 1:
{(3, 2), (4, 3), (5, 4), (6, 5)}
functionA = Function B = Relation
Example 2:
function
Function or relation?
Example 3:
relation
Function or relation?
Example 4:
( x, y) = (student’s name, shirt color)
function
Function or relation?
Example 5: Red Graph
relation
Function or relation?
Example 6
function
Function or relation?
Jacob
Angela
Nick
Greg
Tayla Trevor
Honda
Toyota
Ford
Function Notation
Function form of an equation
• A way to name a function !
• f(x) is a fancy way of writing “y” in an equation. !
• Pronounced “f of x”
Evaluating Functions
8. Evaluating a functionf(x) = 2x – 3 when x = -2
f(-2) = - 4 – 3
f(-2) = - 7
Tell me what you get when x is -2.
f(-2) = 2(-2) – 3
9. Evaluating a functionf(x) = 32(2)x when x = 3
f(3) = 256
Tell me what you get when x is 3.
f(3) = 32(2)3
10. Evaluating a functionf(x) = x2 – 2x + 3 find f(-3)
f(-3) = 9 + 6 + 3
f(-3) = 18
Tell me what you get when x is -3.
f(-3) = (-3)2 – 2(-3) + 3
11. Evaluating a functionf(x) = 3x + 1 find f(3)
f(3) = 28
Tell me what you get when x is 3.
f(3) = 33 + 1
Domain and Range
• Only list repeats once • Put in order from least to
greatest
12. What are the Domain and Range?x 1 2 3 4 5 6y 1 3 6 10 15 21
Domain:
Range:
{1, 2, 3, 4, 5, 6}
{1, 3, 6, 10, 15, 21}
13. What are the Domain and Range?
Domain:
!
!
Range:
{0, 1, 2, 3, 4}
{1, 2, 4, 8, 16}
14. What are the Domain and Range?
Domain:
!
!
Range:
All Reals
All Reals
15. What are the Domain and Range?
Domain:
!
Range:x ≥ -1
All Reals Is Negative -2 in the domain of this function?
Yes : “A” No : “B”
Homework/Classwork
Function Practice
Pg. 76 Skill Practice# 5-7 Pg. 77 # 11,12,28-30 Pg 79 Mixed Practice
#53,55,61
5.1 Rate of Change and Slope
Objective: SWBAT determine the slope of a line given a graph or two
ordered pairs.
Quick PreviewThe diagram at the right shows the side
view of a ski lift.
1. What is the vertical change from:
a. A to B?
b. B to C?
c. C to D?
2. What is the horizontal change from:
a. A to B?
b. B to C?
c. C to D
3. Find the ratio of the vertical change to the horizontal change for each section of the ski lift.
!4. Which section is the steepest?
10 feet
40 feet
10 feet
30 feet
30 feet
30 feet A 1 B 4 C 1 B 3 C 3 D 3
= ; = ; =
B to C
Rate of Change
change of Rate = variabledependent in the changet variableindependen in the change
What is the rate of change for the data to the left?
change of dependent = $15
change of independent = 1 day
$15 = $15/day 1 day !What does the rate of change mean?
It cost $15 more to rent a computer each day.
The Magnum Cedar Point, OH
Ghost Rider Knott’s Berry Farm Xcelerator
Knott’s Berry Farm
Superman 6 Flags Magic
Mountain
Son of Beast King’s Island, OH
How steep is the steepest roller coaster?
These are
completely
vertical!
Imagine a roller coaster moving from LEFT to RIGHT
There are four different kinds of slope the coaster could encounter...
UphillPositiveSlope
Downhill
NegativeSlope
And 2 Special Cases…
The zero-fun coaster
The impossibly
scary coaster
What kind of
slope are they?
Horizontal
Zero
SlopeAw!
That’s zero fun!
Vertical UndefinedSlope
That’s a really scary coaster, so the slope
is…
For the following slides,call out whether the slope is positive, negative, zero, or undefined.
Wait for the count of 3 before you say your answer!
1 2 3negative
1 2 3po
sitive
1 2 3
zero
1 2 3undefined
How do you calculate the slope of a line?
1. Pick two points on the line 2. Calculate “rise over run” 3. Determine the sign
There are 3 steps:
What sign is it?Step
Look to see if the slope is increasingPositive
Or decreasing
Negative
Show me!
Model # 1: Watch me do the three steps
Step 1: Choose 2 points
Rise = 2
Run = 1
RiseRundivided by21
m = 2Step 3:
Determine
the sign
posit
ive
Slope = RiseRun
divided by
Step 2:
+
Practice # 3: Find the slope of this line
RiseRun
divided byRise = 3
Run = 2
23 =m
Think Pair
+Step 1: Choose Two points
Step 2: Rise Over RunStep 3: Determine Sign
Slope Formula
12
12
xxyy
m−
−= 5. Given (5, -3) and (-4, 2), find
the slope of the line.
x1 y1 x2 y2
m =54)3(2
−−
−−
95−
=
Slope Formula
12
12
xxyy
m−
−= 6. Given (0, -2) and (4, -2), find
the slope of the line.
x1 y1 x2 y2
m =04)2(2
−
−−−
40
= 0=
PARALLEL V. PERPENDICULAR
• Parallel lines have the same slope
Parallel v. Perpendicular
• Perpendicular have inverted slope or the inverse.
Wrap Up• Rate of Change (includes units) • Slope !– Positive – Negative – Zero – Undefined !
– Pg. 86 #3,10,11,15 Pg. 87 #41,45
↔runrise �
12
12
xxyy
m−
−=
Graphing Lines Using Slope Intercept Form
Goal: Graph lines in slope intercept form.
Slope-Intercept Form of the Linear Equation
y = mx + bm
= slope
b
= y-interceptAny linear equation which is solved for y
is in slope-intercept form.
Find the slope and y-intercept of the following linear equations:
y = 3x + 4m = 3 b = 4
y = -2x - 1m = -2 b = -1
y = x -9 4
m = b =9 4
-2 1
-2 -1
y = 5x
m = 5 b = 0
+3+2
-3
Graph the following linear equation using slope and y-intercept.
x
y
1) Find the slope and y-intercept.
Steps
2) Plot the y-intercept.
m = 23
or m = -2
4) Draw line through points.
y x= −23
1
32
=m b = −1
3) Plot the slope.-2
-3
+1
-3
Graph the line which passes through (-2, 1) and has a slope of -3.
x
y1) Plot the point.Steps
2) Write slope as fraction
and count off other points.
m = -3 = -31
or m = 3
-1
3) Draw line through points.
+4
+3
Graph the line which passes through (3, 2) and has a slope of .
x
y
1) Plot the point.
Steps
2) Write slope as fraction
and count off other points.
m = 34
or m = -3-4
3) Draw line through points.
34
Write a linear equation in slope-intercept form
to describe each graph. y = mx + b
x
y
x
y
b = 3m =21 y = 2x + 3
b = -431−
=m
y x= − −13
4
4
8
-62
Parallel LinesGraph the following on the coordinate plane.
y x= −12
3 y x= −12
1
m =12
b = −3
m =12
b = −1x
y
Parallel lines have the same slope.
-3x -3x
Tell whether the lines below are parallel.
1) 3x + y = 7 y = -3x + 1
y = -3x + 7
m = -3
y = mx + bm = -3
Lines are parallel!
Same slope!
SKILL PRACTICE !
Pg. 93 #9,11,12,14,23 Pg. 94 #43, 45,47, 53
!
Homework Graphing Worksheet !!
Problems 1-12