FUNCTIONS What are they?. Goal Learn general properties that apply to all types of functions Transform a function so that its graph fits real- world data.

Slides:

Advertisements

Similar presentations

Vocabulary axis of symmetry standard form minimum value maximum value.

Advertisements

Mathematical Models Chapter 2

Chapter 2 Functions and Graphs

State the domain and range of each function. 3.1 Graphs of Exponential Functions.

Excellence Justify the choice of your model by commenting on at least 3 points. Your comments could include the following: a)Relate the solution to the.

Rational Functions Sec. 2.7a. Definition: Rational Functions Let f and g be polynomial functions with g (x ) = 0. Then the function given by is a rational.

1.6 – Inverse Functions and Logarithms. One-To-One Functions A function is one-to-one if no two domain values correspond to the same range value. Algebraically,

EXAMPLE 1 Graph a rational function (m < n) Graph y =. State the domain and range. 6 x SOLUTION The degree of the numerator, 0, is less than the.

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

Barnett/Ziegler/Byleen Finite Mathematics 11e1 Chapter 2 Review Important Terms, Symbols, Concepts 2.1. Functions Point-by-point plotting may be used to.

Copyright © Cengage Learning. All rights reserved. 1 Functions and Limits.

Algebra II Tuesday, September 9, 2014 A Day  Drill: Graph triangle ABC with points A(1, 3), B(5, 6), and C(7, 1). Identify the points of the image of.

exponential functions

Chapter 2 Polynomial and Rational Functions

7-2 Graphing Exponential Functions

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

Chapter 1 Functions and Their Graphs Introduction to Functions Objectives:  Determine whether relations between two variables represent a function.

4.1 Using Graphs to Relate Two Quantities:

Unit 1: Functions 1-1: Key Features of Functions and Real-life Functions.

1.8 Represent Functions as Graphs

SECTION 1.3 PROPERTIES OF FUNCTIONS PROPERTIES OF FUNCTIONS.

Drill #16 Name the set of numbers graphed. Name the set of numbers that each number belongs to Graph the following set on a number line: { -2, 0,

Coordinated Algebra Unit 3 Part B. What is an Exponential Function?

Rational Functions Intro - Chapter 4.4.  Let x = ___ to find y – intercepts A rational function is the _______ of two polynomials RATIO Graphs of Rational.

Properties of Exponential Functions Today’s Objective: I can transform an exponential function.

Let’s see what you remember Find the value of the following.

Area of a circle Radius r (m) Area A (m 2 ) when r = 2.5 A = 20 (to 2sf) when A = 30 r = 3.1 (to 2sf)

Graphs of Motion Physics Ms. Allison. General Graphing Rules Title Label x and y axis with units - Dependent variable is on the y-axis; independent is.

Coordinate Algebra Day 75

Section 1.2 Functions and Graphs. Relation A relation is a correspondence between the first set, called the domain, and a second set, called the range,

WARM UP INVESTIGATION Paper Cup Analysis Find an equation for calculating the height of a stack of paper cups. Instructions: 1.In your groups measure the.

Rational Functions Essential Questions

Lesson 3.6 (Continued) Graphing Exponential Functions : Graphing Exponential Functions.

Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain.

2-7 Absolute Value Function Objective: I can write and graph an absolute value function.

Section 2.2 Graphs of Functions Objectives: Review Domain Find Domain from a graph. Graph piecewise functions.

©2007 by S – Squared, Inc. All Rights Reserved. Description:  b is the base  b > 0 (positive number)  b ≠ 1  a ≠ 0  x is the exponent  x is the.

Math – Exponential Functions

Notes Over 9.2 Graphing a Rational Function The graph of a has the following characteristics. Horizontal asymptotes: center: Then plot 2 points to the.

EXAMPLE 1 Graph y = b for b > 1 x SOLUTION Make a table of values.STEP 1 STEP 2 Plot the points from the table. Graph y =. x 2 STEP 3 Draw, from left to.

WARM UP What is a function. How are they used.. FUNCTIONS.

Chapter 11 Most Missed Topics Velocity and Acceleration.

Chapter 2: Functions and Relations Section 2-3: Functions in the Real World.

A measure of… How far something goes and… How long it takes to get there.

Topic 4 Functions Graphs’ key features: Domain and Range Intercepts

WARM UP What is a function. How are they used.. FUNCTIONS.

Warmup 3-24 Simplify. Show work! Solve for x. Show work! 4. 5.

 “Speed and Velocity”.  Key Questions How do you calculate speed? How can you describe changes in velocity? How can you interpret graphs of distance.

1.4 Modeling with Functions IV

GRAPHING RATIONAL FUNCTIONS

Functions and Their Properties

Warm-up (10 min.) I. Factor the following expressions completely over the real numbers. 3x3 – 15x2 + 18x x4 + x2 – 20 II. Solve algebraically and graphically.

How can you make the arrow point up by moving only three O’s?

Chapter Functions.

Functions: Graphically, Algebraically, Numerically, and Verbally

Do Now: Can you input all real numbers into the x variable in the following functions? If not what numbers can x not take on?

Sullivan Algebra and Trigonometry: Section 6.3 Exponential Functions

EXPONENTIAL FUNCTIONS

Warm-up (10 min.) I. Factor the following expressions completely over the real numbers. 3x3 – 15x2 + 18x x4 + x2 – 20 II. Solve algebraically and graphically.

Graphing Exponential Functions

1.2 Functions and Their Properties

More Properties of Functions

Inverse Functions and Logarithms.

Rational Functions Essential Questions

FUNCTIONS What are they?.

Graphing Exponential Functions

Sullivan Algebra and Trigonometry: Section 6.2

Motion Part 2: Distance-Time Graphs.

Domain The set of all possible input values. Independent Variable X

Graph Rational Functions

Presentation transcript:

FUNCTIONS What are they?

Goal Learn general properties that apply to all types of functions Transform a function so that its graph fits real- world data – Obtain goals Graphically Algebraically Numerically Verbally

Overview (shooting arrow) Graphically  Algebraically: y = −4.9x x + 5 Numerically: x (s)y (m) Verbally: -4.9 is constant pertaining to gravitational acceleration, 20 is initial velocity, and 5 represents initial height of arrow.

Cup of Coffee How does the temperature of a cup of coffee relate to the time it sits??????? There is one and only one temperature at any one given time, sooooooooooo Temperature is a function of time!

Understanding Coffee Algebraically y = (0.8) x. Numerically x (min) y (°C)

Verbal Coffee Mathematical Model – Function that is used to make predictions and interpretations about something in the real world Temperature depends on Time? Or time depend on Temperature? Temp is Dependent Variable Temp depends on time it has been cooling Time is Independent Variable Cannot change time by changing temp!

Domain/Range Domain: The set of values the independent variable of a function can have is called the Domain Range: The set of values of the dependent variable corresponding to the domain is called the range of the function The horizontal line at 20°C is called an asymptote. “Not due to coincide.” End behavior of the graph. ALWAYS PLOT INDEPENDENT VARIABLE ON X-AXIS AND DEPENDENT VARIABLE ON Y- AXIS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Example 1 The time it takes you to get home from a football game is related to how fast you drive. Sketch a reasonable graph showing how this time and speed are related. Give the domain and range of the function.

Final Graph Domain: 0 ≤ speed ≤ speed limit Range: time ≥ minimum time at speed limit