1. Functions and Models

2. AAny set of ordered pairs or any equation that produces sets of ordered pairs is a relation TThe independent variable is given as the first coordinate and the dependent variable is second TThe dependent variable is the one that depends on the independent variable (you may have to determined which is which) AA 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

3.  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)

4. AA linear regression model may or may not go through any of the data points AA linear regression model is a line that estimate the linear relationship between the independent and dependent variable of the data TTo 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

5.  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

6.  Ex1. ◦ A) find a linear regression model ◦ B) find the correlation coefficient ◦ C) Is this a strong or weak correlation? Time12345678910 Money20323136394841454449

7.  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

8.  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 Year191019201927193519391942195619611970 Pop.254631423048425643874571521557135013

9.  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?

10.  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

11.  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?

12. AAll quadratic models are based on quadratic functions of the form f(x) = ax² + bx + c where a ≠ 0 TThe graphs of quadratic models are parabolas IIf a < 0, then the parabola opens down IIf a > 0, then the parabola opens up TTo find the x-intercepts (a.k.a. solutions or zeros), use the quadratic formula RReal world considerations may restrict the domain and/or the range

13. OOpen your book to page 122, we are going to read “Using Known Quadratic Models” YYou need three data points to use your calculator to find a quadratic model (QuadReg) IImpressionistic models or non-theory-based models are when no theory exists that explains why the data fits the model EEx1. 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?

14. IIn a step function, 1 variable will “jump” instead of gradually changing TThe graph of a step function looks like steps TThe greatest integer function is the function f such that for every real number x, f(x) is the greatest integer < x TThe symbol for greatest integer function is TThe greatest integer function is a.k.a. the floor function or the rounding down function IIn your calculator: MATH, NUM, int( ◦Y◦You should be able to do most of these in your head

15. SSolve ◦E◦Ex1.Ex2.Ex3. TThe domain of the greatest integer function is all real numbers TThe range of the greatest integer function is the set of integers SSee the graph of the greatest integer function on page 129 TThe ceiling function or the rounding up function is the smallest integer that > x TThe symbol for the ceiling function is TThe graphing calculator does not have this function, however

16.  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.