Warm-up• Given this relation: {(2, -3), (4, 1), (-5, -3), (-1, 1)

• Domain:

• Range:

• Is it a function? Yes/ No

• Given the graph on the right

• Domain (INQ)

• Domain (INT)

• Range (INQ)

• Range (INT)

Warm-up • Given this relation: {(2, -3), (4, 1), (-5, -3), (-1, 1)

• Domain: {-5, -1, 2, 4}

• Range: {-3, 1}

• Is it a function? Yes/ No

• Given the graph on the right

• Domain (INQ): x ≤ 2

• Domain (INT): (-∞, 2]

• Range (INQ): y ≤ 0

• Range (INT): (-∞, 0]

Sections 2-5 & 8-1

Direct & Inverse Variations

Objectives

• I can recognize and solve direct variation word problems.

• I can recognize and solve inverse variation word problems

Direct Variation

As one variable increases, the other must also increase ( up, up)

ORAs one variable decreases, the other variable must also decrease. (down,

down)

Real life?• With a shoulder partner take a few

minutes to brainstorm real life examples of direct variation. Write them down.

Food intake/weightExercise/weight lossStudy time/ grades

Hourly rate/paycheck sizeStress level/blood pressure

Direct Variation

• y = kx• k is the

constant of variation

• the graph must go through the origin (0,0) and must be linear!!

Direct VariationEx 1)If y varies directly as x and y = 12 when x = 3, find y when x

= 10.

Look for this

key word

Solving Method #1

Use y=kxFIRST: Find your

data points!(x,y)

NEXT: Solve for k & write your equation

LAST: use your “unknown” data point to solve for the missing variable.

OR

yk

x

Solving Method #2

FIRST: Find your data points!

(x,y)

NEXT: substitute your values correctly

LAST: cross multiply to solve for missing variable.

1 2

1 2

y y

x x

You will know

3 of the 4 variables

What did we do?

Use y=kx

FIRST: Find your data points!

(x,y)

NEXT: substitute your values correctly

LAST: cross multiply to solve for missing variable.

FIRST: Find your data points!

(x,y)

NEXT: Solve for k & write your equation

LAST: use your “unknown” data point to solve for the missing variable.

EITHER ONE WILL

WORK!! ITS

YOUR CHOICE!

1 2

1 2

y y

x x

Direct Variation ApplicationEx: In scuba diving the time (t) it takes a diver

to ascend safely to the surface varies directly with the depth (d) of the dive. It takes a minimum of 3 minutes from a safe ascent from 12 feet. Write an equation that relates depth (d) and time (t). Then determine the minimum time for a safe ascent from 1000 feet?

1 2

1 2

d d

t t

2

12 1000

3 t 2

2

12 3000

250 minutes

t

t

Your TURN #3 on Homework

• Find y when x = 6, if y varies directly as x and y = 8 when x = 2.

1 2

1 2

y y

x x 1 8

6 2

y 12 48y

1 24y

Inverse Variation

As one variable increases, the other decreases. (or vice versa)

Inverse Variation• This is a NON-LINEAR

function (it doesn’t look like y=mx+b)

• It doesn’t even get close to (0, 0)

• k is still the constant of variation

Real life?

• With a shoulder partner take a few minutes to brainstorm real life examples of inverse variation. Write them down.

Driving speed and timeDriving speed and gallons of gas in tank

Inverse VariationEx 3) Find y when x = 15, if y

varies inversely as x and when y = 12, x = 10.

Solving Inverse Variation

FIRST: Find your data points!

(x,y)

NEXT: Find the missing constant, k,by using the full set of data given

LAST: Using the formula and constant, k, find the missing value in the problem

x

ky

Method #2

FIRST: Find your data points!

(x,y)

NEXT: substitute your values correctly

LAST: use algebra to solve for missing variable.

1 1 2 2x y x y

You will know

3 of the 4 variables

What did we do?

FIRST: Find your data points!

(x,y)

NEXT: substitute your values correctly

LAST: use algebra to solve for missing variable.

FIRST: Find your data points!

(x,y)

NEXT: Find the missing constant, k,by using the full set of data given

LAST: Using the formula and constant, k, find the missing value in the problem

EITHER ONE WILL

WORK!! ITS

YOUR CHOICE!

x

ky 1 1 2 2x y x y

Inverse Variation Application

Ex:The intensity of a light “I” received from a source varies inversely with the distance “d” from the source. If the light intensity is 10 ft-candles at 21 feet, what is the light intensity at 12 feet? Write your equation first.

1 1 2 2I d I d 210 21 12I

2210 12I17.5 ft-candles

Your TURN #7 on Homework

Find x when y = 5, if y varies inversely as x and x = 6 when y = -18.

1 1 2 2x y x y1 5 6 18x

15 108x 1 21.6x

Direct vs. Inverse Variation

Homework

• WS 1-7

• Quiz next class