Functions and Models
AAny set of ordered pairs or any equation that produces sets of ordered pairs is a relation TThe independent variable is given as the first coordinate and the dependent variable is second TThe dependent variable is the one that depends on the independent variable (you may have to determined which is which) AA vertical line test is used to determine if a relation is a function ◦I◦If it passes the vertical line test it is a function ◦N◦No two y-values can have the same x-value
Write all domains like {x: x < 5} Write all ranges like {y: y > -3} The independent variable is also known as the argument f(a + b) ≠ f(a) + f(b) Ex1. Let f(x) = 4x² - 5, find f(-3)
AA linear regression model may or may not go through any of the data points AA linear regression model is a line that estimate the linear relationship between the independent and dependent variable of the data TTo find the equation for a linear regression model: estimate two points on the line, find the slope between the two points, use the slope and one of the points to find the y-intercept
The correlation coefficient measures how close the data is to being linear A correlation coefficient of 0 means that the data is in no way close to linear The variable for correlation coefficient is r A correlation coefficient of 1 or -1 means that the data is perfectly linear Therefore, 0 < │r│ < 1 and -1 < r < 1 If r > 0, then the correlation is positive If r < 0, then the correlation is negative Make sure your diagnostics are on to find the correlation coefficient
Ex1. ◦ A) find a linear regression model ◦ B) find the correlation coefficient ◦ C) Is this a strong or weak correlation? Time Money
The very best linear regression model is called the line of best fit To find errors in predicted values ◦ Observed value – predicted value ◦ You have to use this order to subtract ◦ See example 1 on page 98 If you are estimating a value that would fall between known data values, that is called interpolation If you are estimating a value that would fall outside of known data values, that is called extrapolation
Extrapolation is a bad idea because of all of the known variables Your calculator finds the line of best fit by finding the sums of the squares of the errors (see the green table on page 98) The center of gravity is the one point on the line of best fit you can determine by hand ◦ The x-coordinate is the mean of the x-values ◦ The y-coordinate is the mean of the y-values Ex1. Find the center of gravity Year Pop
An exponential function with base b is a function with formula of the form y = a·b x where a ≠ 0, b > 0 and b ≠ 1 If b > 1, then it is exponential growth and the graph is an exponential growth curve If 0 < b < 1, then it is exponential decay and the graph is an exponential decay curve Read the properties on page 108 Ex1. An area starts with 20 frogs. The average growth rate is 28%. What is the population at the end of each of the first 3 years?
To determine an exponential model with your calculator you must input at least 2 data points In the graphing calculator, input the data and then choose ExpReg Ex1. Five months after introducing rabbits to an area, there are 128 and after7 total months there are 216. Find the exponential growth model for this situation. The time that it takes a population to double is called the doubling time The time it takes a population to be cut in half is called the half-life
Doubling time and half-life can use any unit of time Ex2. A certain substance has a half-life of 24 years. Initially there were 60 grams of the substance. ◦ A) Write an exponential model for the situation ◦ B) How much will remain in 50 years? ◦ C) When will only 5 grams remain?
AAll quadratic models are based on quadratic functions of the form f(x) = ax² + bx + c where a ≠ 0 TThe graphs of quadratic models are parabolas IIf a < 0, then the parabola opens down IIf a > 0, then the parabola opens up TTo find the x-intercepts (a.k.a. solutions or zeros), use the quadratic formula RReal world considerations may restrict the domain and/or the range
OOpen your book to page 122, we are going to read “Using Known Quadratic Models” YYou need three data points to use your calculator to find a quadratic model (QuadReg) IImpressionistic models or non-theory-based models are when no theory exists that explains why the data fits the model EEx1. A projectile is shot from a tower 10 feet high with an initial upward velocity of 100 feet per second. ◦A◦A) Approximate the quadratic relationship between the height h and time t after the projectile is shot ◦B◦B) How long will the projectile be in the air? ◦C◦C) What is the maximum height of the projectile?
IIn a step function, 1 variable will “jump” instead of gradually changing TThe graph of a step function looks like steps TThe greatest integer function is the function f such that for every real number x, f(x) is the greatest integer < x TThe symbol for greatest integer function is TThe greatest integer function is a.k.a. the floor function or the rounding down function IIn your calculator: MATH, NUM, int( ◦Y◦You should be able to do most of these in your head
SSolve ◦E◦Ex1.Ex2.Ex3. TThe domain of the greatest integer function is all real numbers TThe range of the greatest integer function is the set of integers SSee the graph of the greatest integer function on page 129 TThe ceiling function or the rounding up function is the smallest integer that > x TThe symbol for the ceiling function is TThe graphing calculator does not have this function, however
The graph of the floor function has the first endpoint closed and the second endpoint open The graph of the ceiling function has the first endpoint open and the second endpoint closed Ex4. Ex5.