Fst ch3 notes

3-1 Changing Windows

transformation:

a one-to-one correspondence between sets of points

Name some transformations

translationreflectionrotation


asymptote (a-simp-tote):

a line that the curve approaches but never touches

points of discontinuity:

points where there is a break in the graph


automatic grapher:

Our graphing calculator. This is a function grapher.

default window:

The part of the coordinate plane that shows on the screen of the calculator.

viewing window:

viewing rectangle:

−10 ≤ x ≤ 10 and −10 ≤ y ≤ 10


parent function:

a simple form or the simplest form of a class of functions from which other members of the class can be derived by transformations.

Examples of parent functions: y =x

y=x2

y=x3

y= x

y= x

3-1 Examples

1. Sketch the graph of .

y =10 −x2

Get specific points to plot using the table feature on your calculator

x y

-5 -15

-4 -6

-3 1

-2 6

-1 9

0 10

1 9

2 6

3 1

4 -6

5 -15

Get specific points to plot using the table feature on your calculator

x y

-5 3750

-4 1664

-3 594

-2 144

-1 14

0 0

1 -6

2 -16

3 54

4 384

3-1 Examples

2. Sketch the graph of .

y =4x4 −10x3

3-2 The Graph Translation Theorem

translation image:

the result of a translation

preimage:

the original image

3-2 The Graph Translation Theorem

translation (of a graph):

a transformation that maps each point (x,y) onto (x+h, y+k)

Graph Translation Theorem:

In a relation described by a sentence in x and y, the following two processes yield the same graph:

1. replacing x by x-h and y by y-k in the sentence;

2. applying the translation (x,y) onto (x+h, y+k) to the

graph of the original relation.

3-2 Examples

1. Under a translation, the image of (0, 0) is (7, 8).

a. Find a rule for this translation.

b. Find the image of (6, -10) under this translation.

(x, y) → (x+7, y+8)

(x, y) → (x+h, y+k)Think: what did you do to “0” to get to “7”?

You added “7”

Think: what did you do to “0” to get to “8”?You added “8”

This is the Rule!

(x, y) → (x + 7, y + 8) Start with the rule

Substitute into the rule(6,-10) → (6 + 7, -10 + 8)

(6,-10) → (13, -2) Simplify

3-2 Examples

2. Compare the graphs of y=x3 and y + 5=(x + 4.2)3.

I like to think of the equation as y=(x + 4.2)3 - 5.

This graph is a translation of y=x3 by shifting it left 4.2 and down 5.

3-2 Examples

2. Consider the graphs of y=x3 and y=(x + 4.2)3 - 5. Find the rule for translating (x, y) → (x + h, y + k).

y=(x + 4.2)3 - 5

y=(x - h)3 + k

y=(x - -4.2)3 + -5

h= -4.2, k= -5

(x, y) → (x - 4.2, y - 5)

Hint: find h and k from the equation

Find the coordinates of (1, 1) on the translated graph.

(1, 1) → (1 - 4.2, 1 - 5)

(1, 1) → (-3.2, -4)

3-2 Examples

3. If the graph of y=x2 is translated 2 units up and 3 units to the left, what is an equation for its image?

y=x2 y=(x-h)2 +k

That’s “h”That’s “k”k=2 h=-3

positive negative

y=(x-(-3))2 +2

y=(x+3)2 +2

3-3 Translations of Data

translation (of data):

a transformation that maps each xi to xi + h where h is some constant

T : x → x+h or T(x) =x+h

translation image (of a data value):

the result of a translation

invariant:

does not change

3-3 Translations of Data

Theorem:

Adding “h” to each number in a data set adds “h” to each of the mean, median, and mode.

Theorem:

Adding “h” to each number in a data set does not change the range, IQR, variance, or standard deviation of the data.

invariant

3-3 Examples

1. Ten students earned the following scores on a test: 93, 95, 91, 96, 88, 90, 93, 95, 80, 100. Translate the data mentally by subtracting 90 to mentally find the mean of these scores.

Think: 10 scores; get total by adding over / under 90.

i.e. 93 is over 90 by three, so think +395 is over 90 by 5, so think +5 (total of +8)

continue this for all of the data and mentally compute the total

You should have a total of +21. Now, compute the mean.

x =2110

=2.1

Then add 90 to the mean x =2.1+ 90 =92.1

3-3 Examples

1. A worker records the time it takes to get from home to the parking lot of the factory and finds a mean time of 20.6 minutes with a standard deviation of 3.5 minutes. If it consistently takes 5 minutes to get from the parking lot to the worker’s place in the factory, find the mean and standard deviation of the time it takes the worker to get from home to that place in the factory.

mean = 20.6 + 5mean = 25.6 minutes

standard deviation is not affected (invariant), therefore the standard deviation remains 3.5 minutes

What other statistics will change?What other statistics will remain invariant?

3-5 The Graph Scale-Change Theorem

vertical scale change, stretch:

vertical scale factor:

A transformation that maps (x,y) to (x, by)

The number “b” in the transformation that maps (x,y) to (ax, by)


horizontal scale change, stretch:

horizontal scale factor:

A transformation that maps (x,y) to (ax,y)

The number “a” in the transformation that maps (x,y) to (ax,by)


scale change (of a graph):

size change:

The transformation that maps (x,y) onto (ax,by)

A scale change in which the scale factors (a and b) are equal


Graph Scale-Change TheoremIn a relation described by a sentence in x and y,

replacing x by x

a and y by

y

b in the sentence yields

the same graph as applying the scale change

(x, y)→ (ax,by) to the graph of the original relation.Yikes!

1. Replace x with x

a and y with

y

b.

2. Apply the transformation (x, y)→ (ax,by)to the graph of the original relation.

Equation

Points


Before we continue, we need to practice!

x1

3

=3xx1

5

=5x

7x = x1

7

6x = x1

6

11x = x1

11

x12

=2x

3-5 Examples

1. Compare the graphs of y = x and y= 6x .

y = x

y = 6x

Hint: The graph of y = 6x is the image

of y=x under a horizontal scale change of

magnitude 16.

3-5 Examples

2. Sketch the graph of y

4= 6x .

Hint: The graph is the of y

4= 6x is the image

of y=6x under a vertical scale change of

magnitude 4. y =4 6x

y = x

3-5 Examples

3. Sketch the image of y =x3 under S(x,y) =(−2x,y).

S(x, y) =(−2x,y)S(−2,−8) =(4,−8)S(−1,−1) =(2,−1)

S(0,0) =(0,0)S(1,1) =(−2,1)

S(2,8) =(−4,8)

y =x3y =−

18

x3

3-5 Examples

3. b. Give an equation of the image of

y =x3 under S(x,y) =(−2x,y).

Hint: replace "x" with "x

a" and "y" with "

y

b"

y =x3

y

b=

xa

⎛⎝⎜

⎞⎠⎟

3

y

1=

x−2

⎛⎝⎜

⎞⎠⎟

3

y =−18

x3

3-6 Scale Changes of Data

scale factor:scale image:

The number “a” in the scale change

scale change (of data):

a transformation that maps each xi to axi where a is some non-zero constant. That is, S is a scale change iff

S : x → ax or S(x) =ax

scaling:rescaling:

When a scale change is applied to a data set

3-6 Scale Changes of Data

Theorem:

Multiplying each element of a data set by the factor “a” multiplies each of the mean, median, and mode by the factor “a”.

Theorem:

If each element of a data set is multiplied by “a”, then the variance is “a2” times the original variance, the standard deviation is |a| times the original standard deviation, and the range is |a| times the original range.

In other words: everything get multiplied by “a” (except the variance which is “a2”)

3-6 Examples

1. The teachers in a school have a mean salary of $30,000 with a standard deviation of $4,000. If each teacher is given a 5% raise, what will be their new mean salary, and what will be their new standard deviation?

Original Mean New Mean

Original Standard Deviation New Standard Deviation

$30,000 $30,000 (1.05)

$31,500

$4,000 $4,000 (1.05)

$4,200

3-6 Examples

2. To give an approximate conversion from miles to kilometers, you can multiply the number of miles by 1.61. Suppose data are collected about the number of miles that cars can go on a tank of gas. What will be the effect of changing from miles to kilometers on:

a. the median of the data?

c. the standard deviation of the data?

b. the variance of the data?

multiplied by 1.61

multiplied by (1.61)2

multiplied by 1.61

3-8 Inverse Functions

Horizontal Line Test:the inverse of a function f is itself a function iff no horizontal line intersects the graph of f in more than one point.

inverse of a function:

the relation formed by switching the coordinates of the ordered pairs of a given function (switch x and y)

inverse function, f -1:

notation for the inverse of a function

3-8 Inverse Functions

identity function:

I(x) = x

Inverse Function Theorem:

Any two functions f and g are inverse functions ifff(g(x))=x and g(f(x))=x

one-to-one function:

a function in which no two domain values correspond to the same range value.

3-8 Examples

1. a. Find the inverse of S = {(1,1), (2,4), (3,9), (4,16)}.

1. b. Describe S and it’s inverse in words.

S−1 = (1,1),(4,2),(9, 3),(16,4){ }

S is the squaring function.S-1 is the square root function.

3-8 Examples

In 2 and 3, give an equation for the inverse of the function and tell whether the inverse is a function.

2. f (x) =6x+ 5

y =6x+ 5

x =6y+ 5

x −5 =6y1

6x −

56=y

f −1(x) =16

x−56

Function!

3. y =4

3x−1

x =4

3y−1

x(3y −1) =4

(3y −1) =4x

3y =4x+1 y =

43x

+13

Function!

3-9 z-scores

z-score:

z =x−x

s

Suppose a data set has a mean x and standard deviation s.

The z-score for a member x of this data set is

raw score:

the original data

raw data:

3-9 z-scores

the results of a transformation

standardized scores:

standardized data:

3-9 z-scores

Theorem:

If a data set has a mean x and standard deviation s,

the mean of its z-scores will be 0, and the standard

deviation of its z-scores will be 1.

3-9 Examples

1. Julie took a test at the fourth month of 6th grade. The mean score of students taking this test is theoretically 6.4, and the standard deviation is 1.0. Julie scored 7.8. What is her z-score?

z =x−x

s

z =7.8 −6.4

1

z =1.4

3-9 Examples2. Melvin scored 83 on a test with a mean of 90 and a standard deviation of 6. He scored 37 on a test with a mean of 45 and a standard deviation of 5. On which test did he score in a lower percentile?

z =x−x

sz =

x−xs

z =83−90

6z =

37 −455

z =−1.17 z =−1.6

Test #1 Test #2Which is the

lower percentile?

Education

Fst ch3 notes