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y = ¾ x. Regents Review #2. f(x) = 2x – 5. Linear & Exponential. Functions. g(x) = (½) x. f(x) = 1.5(4) x. Functions. Which relation represents a Function?. What is a function? A relation in which every x-value(input) is assigned to exactly one y-value (output). 6. 2. 7. Function.

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regents review 2

y = ¾x

Regents Review #2

f(x) = 2x – 5

Linear & Exponential

Functions

g(x) = (½)x

f(x) = 1.5(4)x

functions

Functions

Which relation represents a Function?

What is a function?

A relation in which every x-value(input) is assigned to exactly one y-value (output)

6

2

7

Function

Not a Function

functions1

Functions

Which graph represents a function?

We can recognize functions using the vertical line test

Vertical Line Test: If a graph intersects a vertical line in more than one place, the graph is not a function

Function

Not a function

functions2
Functions

Functions can be written using function notation

“f(x)” is read f of x

Example: f(x) = 2x – 3 is the same as y = 2x – 3

x: input

f(x): output

Evaluating Functions: Find f(-10)

f(-10) = 2(-10) – 3

f(-10) = -20 – 3

f(-10) = -23 (-10, -23)

linear functions
Linear Functions

Linear Functions “y = mx +b”

The easiest ways to graph a linear function are…

  • Table of Values

2) Slope-Intercept Method

linear functions1
Linear Functions

Table of Values Method

Graph 2x – 4y = 12 y = ½ x – 3

2x – 4y = 12

Domain: {x|x is all Real #s}

Range: {y|y is all Real #s}

Positive Slope

linear functions2
Linear Functions

Slope-Intercept Method y = mx + b

m = slope

b = y –intercept (0,b)

Graph 6x + 3y = 9

y = -2x + 3

m =

b = 3 (0, 3)

6x + 3y = 9

Domain: {x|x is all Real #s}

Range: {y|y is all Real #s}

Negative Slope

linear functions3
Linear Functions

Vertical Lines

x = a where a represents the x-intercept

x = 4 (undefined slope)

Horizontal Lines

y = b where b represents the y-intercept

y = 4 (zero slope)

y = 4

x = 4

Domain: all real #’s Range: y|y = 4

Domain: x|x = 4 Range: all real #’s

linear functions4
Linear Functions

Writing the Equation of a Line

Write the equation of a line that runs through the points (-3,1) and (0,-1)

Find the slope (m)

(-3,1) (0,-1)

Find the y-intercept (b)

y = mx + b Pt.(-3,1)

1 = (-2/3)(-3) + b

1 = 2 + b

-1 = b

Write the equation in “y = mx + b”

y = x – 1

b = -1

m = -2/3

linear functions5
Linear Functions

Write the equation of a line that is parallel to y – 2x = 4 and runs through the point (-2,4)

Find the slope

Parallel lines have the same slope

y – 2x = 4 y = 2x + 4

m = 2

Find the y-intercept

y = mx + b Pt.(-2,4)

4 = 2(-2) + b

4 = -4 + b

8 = b

b = 8

Write the equation in “y = mx + b”

y = 2x + 8

linear functions6
Linear Functions

The graph shows yearly cost based on the number of golf games played at a private club. Write an equation that represents the relationship shown.

y-int: (0, 90) $90 initial fee

slope (rate of change):

$30 per game

y = 30x + 90

x: # of golf games

y: total cost

(3,180)

(2,150)

linear functions7
Linear Functions

Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 calories. Graph the function, C, where C(x) represents the number of calories in x mints.

modeling data with functions
Modeling Data with Functions

Scatter Plots: A graph of plotted points that show the relationship between two sets of data.

Correlation Coefficient (r): A number in between -1 and 1 that describes the strength of the data.

Calculator :

2nd 0 (CATALOG)

Scroll down to DIAGNOSTICS ON

ENTER, ENTER

modeling data with functions1
Modeling Data with Functions

The local ice cream shop keeps track of how much ice cream they sell as compared to the noon temperature on that day. Here are their figures for the last 12 days:

Regression Line

Trend Line

Line of Best Fit

Least Squares Line

modeling data with functions2
Modeling Data with Functions

Interpolation is where we find a value inside our set of data points.

Here we use  interpolation to estimate the sales at 21 °C.

Extrapolation is where we find a value outside our set of data points.

Here we use extrapolation to estimate the sales at 29 °C (which is higher than any value we have).

modeling data with functions3
Modeling Data with Functions

Write the regression equation (y = ax + b) for the raw score based on the hours tutored. Round all values to the nearest hundredth.

Equation: y = 6.32x + 22.43

x: # of hours tutored

y: raw test score

1) STAT Edit (#1)

2) Enter data

into L1 and L2

3) STAT CALC

LinReg(ax + b)

Using the regression equation, predict the score of a student who was tutored for 3 hours.

y = 6.32(3) + 22.43

y = 41.39

Predicted Raw Test Score: 41

modeling data with functions4
Modeling Data with Functions

Calculating Residuals: A residual is calculated by finding the difference between the actual data value and the predicted value (Actual – Predicted).

The actual score is about 6 points below what I would expect after 3 hours of tutoring.

y = 6.32(3) + 22.43

y = 41.39

Resdiual: 35 – 41.39 = -6.39

A–P = – 6.4

exponential functions
Exponential Functions

There are two types of Exponential Functions

  • Exponential Growth

y = abx where b > 1

  • Exponential Decay

y = abx where 0 < b < 1

Rate of Change is NOT Constant.

An average rate of change can be calculated over a specified interval (see study guide for example).

exponential functions1
Exponential Functions

Domain: All real Numbers {x|x is all Real Numbers}

Range: All real numbers greater than 0 {y|y > 0}

The function is increasing (x and y both increase)

exponential functions2
Exponential Functions

Domain: All real Numbers {x|x is all Real Numbers}

Range: All real numbers greater than 0 {y|y > 0}

The function is decreasing (x increases and y decreases)

exponential functions3
Exponential Functions

What happens to f(x) = 2x when….

4 is added multiplied by -1

f(x) = 2x + 4 f(x) = -2x

Moves f(x) = 2x up 4 units

Reflects f(x) = 2x in the x-axis

exponential functions4
Exponential Functions

Exponential Growth Model y = a(1 + r)t

The cost of maintenance on an automobile increases each year by 8%. If Alberto paid $400 this year for maintenance for his car, what will the cost be (to the nearest dollar) seven years from now?

a: initial value

r:growth rate

t: time

1 + r:growth factor

y = a(1 + r)t

y = 400(1 + .08)7

y = 400(1.08)7

y = 685.5297…

The cost will be $686.00

exponential functions5
Exponential Functions

Exponential Decay Model y = a(1 – r)t

A used car was purchased in July 1999 for $12,900. If the car loses 14% of its value each year, what was the value of the car (to the nearest penny) in July 2003?

a: initial value

r:decay rate

t: time

1 – r:decay factor

y = a(1 – r)t

y = 12,900(1 – .14)4

y = 12,900(.86)4

y = 7056.4052…

The cost of the car was $7056.41

sequences
Sequences

A sequence is an ordered list of numbers.

Use these formulas to define sequences and find the nth term of any sequence.

Arithmetic: an = a1 + d(n – 1)

Geometric: an = a1rn – 1

a1 : first term in the sequence

d: common difference ( + )

r: common ratio (x)

sequences1
Sequences

The first row of the theater has 15 seats in it. Each subsequent row has 3 more seats than the previous row.

Write an explicit formula to find the number of seats in the nth row.

Arithmetic:

Find the number of seats in the tenth row.

sequences2
Sequences

Brian has 2 parents, 4 grandparents, 8 great-grandparents and so on.

Write an explicit formula for the number of ancestors Brian has in a generation if he goes back to the nth generation.

Geometric:

Find the number of ancestors in the 7th generation.

slide27

Now it’s your turn to review on your own! Using the information presented today and the study guide posted on halgebra.org, complete the practice problem set.Regents Review #3 Friday, May 16thBE THERE!