~ Chapter 5 ~Solving & Applying

Proportions

Algebra I

Lesson 5-1 Relating Graphs to Events

Lesson 5-2 Relations & Functions

Lesson 5-3 Function Rules, Tables & Graphs

Lesson 5-4 Writing a Function Rule

Lesson 5-5 Direct Variation

Lesson 5-6 Describing Number Patterns

Chapter Review

Algebra I

~ Chapter 5 ~Cumulative Review Ch 1-

4

Algebra I Algebra I

Relating Graphs to EventsNotesLesson 5-1

Interpreting Graphs

You can use a graph to show the relationship between two variables…

What does each section of the graph represent?

Sketching a Graph

A plane is flying from NY to London. Sketch a graph of the planes altitude during the flight. Label each section…

Relating Graphs to EventsNotesLesson 5-1

Sketch a graph of the distance from a child’s feet to the ground as the child jumps rope. Label each section.

Analyzing Graphs

A car travels at a steady speed.

Which graph could you use?

CA B

Relating Graphs to Events

HomeworkLesson 5-1

Homework ~ Practice 5-1

Relations & Functions Practice 5-1

Lesson 5-2

Relations & Functions Practice 5-1

Lesson 5-2

Relations & Functions NotesLesson 5-2

Relation – a set of ordered pairs

The domain of a relation is the first set of coordinates (x values)

The range of a relation is the second set of coordinates (y values)

Find the domain & range of the relation represented by the data in the table.

{-2,-1,4}{-2,1,3} list in order from least to greatest.

A function [f(x)] is a relation that assigns exactly one value in the range to each value in the domain.

One way to tell whether a relation is a function is to use the vertical-line test. If a vertical line passes through more than one point… the relation is NOT a function.

Using a mapping diagram

If the domain maps to more than one range… then the relation is not a function. If the domain only maps to one range then the relation is a function.

Relations & Functions NotesLesson 5-2

A function rule is an equation that describes a function.(Input – x, output – y)

Evaluating a function rule…

For x = 2.1

y = 2x + 1 f(x) = x2 – 4 g(x) = -x + 2

y = 5.2 f(2.1) = 0.41 g(2.1) = -0.1

Finding the Range

You can use a function rule and a given domain to find the range of the function…

Find the range of each function for the domain {-2,0,5}

f(x) = x – 6

Range = {-8,-6,-1}

y = -4x

Range = {-20,0,8}

g(t) = t2 + 1 y = ¼ x

Relations & Functions

HomeworkLesson 5-2

Homework ~ Practice 5-2

Function Rules, Tables & Graphs NotesLesson 5-3

Independent variable – x – the inputs are values for this variable.

Dependent variable – y – the outputs are values for this variable.

The independent variable graphs on the x-axis, the dependent variable graphs on the y-axis.

Model the function rule y = ½ x + 3 using a table of values and a graph…

To make a table, choose input values for x and evaluate to find y.

To graph, plot points for the ordered pairs from your table… (x,y)

Join the points to form a line or a curve.

Your turn… Model the function rule f(x) = 3x + 4

A recording company charges $300 for making a master CD and designing the art. It charges $2.50 for burning each CD. Use the function rule P(c) = 300 + 2.5 c. Make a table of values and a graph.

Graphing functions

Graph the function y = |x| + 1

(hint: make a table of values and then graph)

Function Rules, Tables & Graphs NotesLesson 5-3

Graph the function f(x) = x2 - 1

Homework ~ Practice 5-3

Writing a Function Rule Practice 5-2

Lesson 5-4

Writing a Function Rule

Practice 5-3Lesson 5-4

Writing a Function Rule

Practice 5-3Lesson 5-4

Practice 5-3

Practice 5-3

Writing a Function Rule NotesLesson 5-4

Writing a rule from a table…

Ask what can be done to 1 to get to -1… Then see if that rule

applies to get from 2 to 0. If not then try again…

So f(x) = x – 2

Rule?

y = 2x

Write a function rule to calculate the cost of buying apples at$1.25 a pound.

Write a function rule to calculate the total distance d(n) traveled after n hours at a constant speed of 45 miles per hour.

Write a function rule to calculate the area A(r) of a circle with radius r.

x f(x)

1 -1

2 0

3 1

4 2

x y

1 2

2 4

3 6

4 8

x f(x)

1 1

3 9

6 36

9 81

x f(x)

-8 64

-4 16

0 0

4 16

8 64

Writing a Function Rule

HomeworkLesson 5-4

Homework – Practice 5-4

Direct Variation

Practice 5-4Lesson 5-5

Direct VariationNotesLesson 5-5

Direct variation – a function in the form ~ y = kx, where k ≠ 0. x & y vary directly… meaning that if x increases in value, y increases in value, and vice versa.

The constant of variation, k, is the coefficient of x.

Determine if an equation is a direct variation…

5x + 2y = 0 Solve the equation for y

2y = -5x

y = -5/2 x (this is in the form y = kx so 5x + 2y = 0 is a direct variation)

What is k in 5x + 2y = 0?

k = -5/2

? 7y = 2x 3y + 4x = 8 y – 7.5x = 0

Writing an equation given a point

Write an equation of the direct variation that includes the point (4, -3)

Remember… y = kx substitute ~ so -3 = k(4) and solve for k

k = -3/4 so the equation is y = -3/4 x

Direct VariationNotesLesson 5-5

Write an equation of the direct variation that includes the point (-3, -6)

y = kx… -6 = k(-3)

k = 6/3 = 2 so… y = 2x

A recipe for one dozen muffins calls for 1 cup of flour. The number of muffins varies directly with the amount of flour you use. Write a direct variation for the relationship between the number of cups of flour and the number of muffins.

x = 1 y = 12 so… 12 = k(1) so k = 12

y = 12x

Direct Variations & Tables

You can rewrite a direct variation y = kx as y/x = k.

For each table, use the ratio y/x, to determine whether y varies directly with x.

In a direct variation, the ratio is the same for all pairs of data where x ≠ 0.

So the proportion x1/y1 = x2/y2 for (x1 ,y1) & (x2, y2)

x y

-2 3.2

1 2.4

4 1.6

x y

4 6

8 12

10 15

Direct Variation

HomeworkLesson 5-5

Homework – Practice 5-5 odd

Describing Number Patterns

Practice 5-5Lesson 5-6

Describing Number Patterns

NotesLesson 5-6

Extending number patterns… What are the next two numbers in the pattern?

1, 4, 9, 16, … rule?

3, 9, 27, 81, … rule?

2, -4, 8, -16, … rule?

Each number in a sequence is a term.

The common difference (d) is a fixed number in a sequence that is added to each previous term resulting in the next term in the sequence.

Determine the common difference in each sequence…

11, 23, 35, 47, …

8, 3, -2, -7, …

In a sequence the term is considered to be the output (y). The input(x) is the number of the term in the sequence.

Arithmetic sequence = A(n) = a + (n-1)d, where n = term number, a = first term, and d = common difference.

Describing Number Patterns

NotesLesson 5-6

Write the arithmetic sequence for 18, 7, -4, -15,…

a = d =

so A(n) = + (n - 1)

Use the equation to find the 10th term in the sequence above.

A(10) = 18 + (10-1)(-11)

A(10) = 18 + (-99) A(10) = -81

Your turn.. 1/2 , 1/3, 1/6, 0, … find the 12th term in the sequence.

d = - 1/6 so A(n) = 1/2 + (n – 1) (-1/6) and A(12) = 1/2+(12 – 1) (-1/6)

A(12) = 1/2+(-11/6) = 3/6 + (-11/6) = -8/6 = -4/3 = -1 1/3

Given the Arithmetic Sequence, find the 3rd, 5th, & 8th term…

A(n) = -2.1 + (n – 1)(-5) (plug and chug…)

A(3) = -2.1 + (3 – 1)(-5)

A(5) = -2.1 + (5 – 1)(-5)

A(8) = -2.1 + (8 – 1)(-5)

18 -11

18 (-11)

A(3) = -2.1 + (-10) = -12.1

A(5) = -2.1 + (-20) = -22.1

A(8) = -2.1 + (-35) = -37.1

Describing Number Patterns

HomeworkLesson 5-6

Homework – Practice 5-6 even

Chapter 5 Review due tomorrow

Describing Number Patterns

Practice 5-6Lesson 5-6

~ Chapter 5 ~Chapter Review

Algebra I Algebra I

~ Chapter 5 ~Chapter Review

Algebra I Algebra I

~ Chapter 5 ~Chapter Review

Algebra I Algebra I