Direct and Inverse.

Slides:



Advertisements
Similar presentations
A3 3.7 Direct and Indirect Variation
Advertisements

3.4-1 Variation. Many natural (physical) phenomena exhibit variation = one quantity (quantities) changing on account of another (or several) Principle.
a.k.a. Proportion functions
Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such th at 3.7 – Variation The number k is.
The general equation for DIRECT VARIATION is k is called the constant of variation. We will do an example together.
2.6 Scatter Diagrams. Scatter Diagrams A relation is a correspondence between two sets X is the independent variable Y is the dependent variable The purpose.
Direct and Inverse Variations Direct Variation Which of these tables models direct variation? If so, write an equation. NO.
Direct and Inverse Variations Direct Variation When we talk about a direct variation, we are talking about a relationship where as x increases, y increases.
Direct and Inverse Variation
Direct and Inverse Variations Direct Variation When we talk about a direct variation, we are talking about a relationship where as x increases, y increases.
Copyright © Cengage Learning. All rights reserved. Graphs; Equations of Lines; Functions; Variation 3.
Warm up Determine the asymptotes for: 1. x=-2, x=0, y=1.
Mathematical Modeling & Variation MATH Precalculus S. Rook.
Direct and Inverse Variations Direct Variation When we talk about a direct variation, we are talking about a relationship where as x increases, y increases.
Certain situations exist where:  If one quantity increases, the other quantity also increases.  If one quantity increases, the other quantity decreases.
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: Developmental.
Direct Variation  Let x and y denote two quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero number.
Direct and Inverse.
2.7 Variation. Direct Variation Let x and y denote 2 quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero.
1.11 Making Models Using Variation. 2 Objectives ► Direct Variation ► Inverse Variation ► Joint Variation.
Direct Variation What is it and how do I know when I see it?
Direct and Inverse Variations Do now: if 5 boxes of salt costs 15 dollars, how much does 4 boxes cost?
UNIT 2, LESSON 8 VARIATION. THREE TYPES OF VARIATION.
k is called the constant of variation or constant of proportionality.
Joint and Combined Variation Review of Variations Direct Variation Inverse Variation Formula General Equation.
Unit 8: Day 1 Direct and Inverse Variation. Definition… Direct Variation: y varies directly as x This means as x increases, y __________ as x decreases,
7.3 Ratio, Proportion, and Variation Part 2: Direct and Indirect Variation.
Direct, Inverse & Joint Variation Section 2.5. Direct Variation 2 variables X & Y show direct variation provided y = kx & k ≠ 0. The constant k is called.
Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
how one quantity varies in relation to another quantity
Direct and Inverse Variations
Direct and Inverse.
Linear Functions and Equations
NOTES 1-1C & D: PROPERTIES DIRECT & INVERSE (INDIRECT) VARIATION
9.1 Inverse & Joint Variation
Mathematical Relationships
Lesson 6-9: Variation Objective: Students will:
Variation Objectives: Construct a Model using Direct Variation
Inverse & Joint Variation
Model Inverse and Joint Variation
Rational Expressions and Functions
Warm-up b) a) Solve the following equation.
2.4 More Modeling with Functions
Graphing.
Rational Expressions and Functions
Direct and Inverse.
Inverse Variations Unit 4 Day 8.
Direct and Inverse Variations
Direct and Inverse VARIATION Section 8.1.
Direct and Inverse Variations
8.1 Model Inverse & Joint Variation
Atmospheric Pressure Pressure is equal to a force per area. The gases in the air exert a pressure called atmospheric pressure. Atmospheric pressure is.
2.5 Model Direct Variation
Direct and Inverse Variations
Lesson 5-2 Direct Variation
Direct and Inverse.
8-5 Variation Functions Recognize and solve direct and joint variation problems. Recognize and solve inverse and combined variation problems.
2.4 More Modeling with Functions
Work Done by a Varying Force
Direct and Inverse.
Copyright © Cengage Learning. All rights reserved.
Direct and Inverse.
Mathematical Relationships
Atmospheric Pressure Pressure is equal to a force per area. The gases in the air exert a pressure called atmospheric pressure. Atmospheric pressure is.
Direct Inverse and VARIATION.
Direct Variation Two types of mathematical models occur so often that they are given special names. The first is called direct variation and occurs when.
Inverse.
What is it and how do I know when I see it?
Topic: Inverse Variation HW: Worksheet
Model Inverse and Joint Variation
Presentation transcript:

Direct and Inverse

VARIATION

The general equation for DIRECT VARIATION is k is called the constant of variation. We will do an example together.

Direct Variation When we talk about a direct variation, we are talking about a relationship where as x increases, y increases or decreases at a CONSTANT RATE.

If y varies directly as x, and y=24 and x=3 find y when x=2. (a) First, you will need to find the constant of variation (k). Write the general equation Substitute-you know y=24 when x=3.

(b) Find y when x=2 First we find the constant of variation, which was k=8 Now we substitute into y=kx.

Now, YOU try one!! If y varies directly as x and y=6 when x=5, then find y when x=15.

Now lets solve using the equation.

Now try another on your own. y varies directly as x, and x=8 when y=9. Find y when x=12. Answer: 13.5

What does the graph y=kx look like? A straight line with a y-intercept of 0.

Looking at the graph, what is the slope of the line? Answer: 3 Looking at the equation, what is the constant of variation? Answer: 3 The constant of variation and the slope are the same!!!!

We will apply what we know and try this problem. According to Hook’s Law, the force F required to stretch a spring x units beyond its natural length varies directly as x. A force of 30 pounds stretches a certain spring 5 inches. Find how far the spring is stretched by a 50 pound weight.

Set up an equation. Substitute Force = k(length stretched) 30 = k(5)

Now try this problem. Use Hook’s Law to find how many pounds of force are needed to stretch a spring 15 inches if it takes 18 pounds to stretch it 13.5 inches. Answer: 20 pounds

Inverse Variation Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down. There is not necessarily a constant rate.

Inverse Variation y varies inversely as x if such that

Inverse Variation With Direct variation we Multiply our k’s and x’s. y=kx In Inverse variation we will Divide them. y=k/x

Find y when x=15, if y varies inversely as x and x=10 when y=12 Solve by equation:

Solve this problem on your own! Find x when y=27, if y varies inversely as x and x=9 when y=45. Answer: 15

Lets apply what we have learned. The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

Step #1: Set up an equation.

Now try this one on your own. A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

What does the graph of xy=k look like? Let k=5 and graph.

This is a graph of a hyperbola. Notice: That in the graph, as the x values increase the y values decrease. also As the x values decrease the y values increase.

Joint Variation is when one quantity varies directly as the product of two or more quantities. So, just like direct, but with more than one variable. y=kxz

Combined is when you use more than one variation in one problem. For example if y varies directly with x and inversely with z, it would be set up as y=(kx)/z