On this page I will post some notes and examples about what we cover each day in class. I will try and keep this up to date as much as possible. Consider it a mini text book to refer to.
Intro to Functions:
A function is a rule that assigns each element of the domain to a single element of the range. Domain is the x values, the input, and the independent variable. The Range is the y values, the output, and the dependent variable.
A mapping shows a function if there is only one arrow leaving each number in the domain or the first circle.(It doesn't matter if multiple arrows go into the same number in the range.) A mapping that is not a function has more than one arrow leaving a single number in the domain where those arrows go to different numbers in the range.
A graph shows a function if it passes the vertical line test. The vertical line test says that if I draw a vertical line at any point on the graph the vertical line will only intersect the function at one point.
Graphing Systems of Equations and Inequalities
The easiest way to graph an equation is to put it into slope intercept form (y=mx+b).
Graphing inequalaties. If the inequality is less than or greater than, then the line is dotted. If the inequality is less than or equal to or greater than or equal to then the line is a solid line. To decide which side of the line to shade, plug a point on one side of the line ( the point (0,0) is always easy) into the equation and see if the inequality is true. If it is true then shade on that side of the line. If the inequality is false then shade the side of the line opposite the point you chose.
Solving Systems Graphically and with Tables
There are three types of solutions:
Consistent and Independent: There is one solution. The graph has two lines that intersect at a single point. The table has one value of X where Y1 and Y2 are the same value.
Consistent and Dependent: There is infinetly many solutions or all solutions. The graph has two lines on top of each other so it looks like a single line. The table has Y1 and Y2 the same value for every X.
Inconsistent: There is no solution. The graph has two parallel lines. The table has Y1 and Y2 different for every value of X.
Writing Systems from Story Problems
There are no hard fast rules that work for every situation. Some tips are define your variables for the situation and look for the two or more different ideas that they are talking about and write an equation/inequality for each idea. Other than that it just takes practice.
A sequence is a series of numbers, called terms, that are in a particular order.
To write a function for a sequence here are some rules. If the pattern in the sequence is that you add the same number over and over then the function is going to contain that number multiplied by n. If the pattern in the sequence is that you multiply by the same number over and over then the function is going to contain that number to the nth power. If the pattern switches between positive and negative you can multiply your function by (-1)^n for patterns that start negative or (-1)^(n+1) for patterns that start positive.
To find the nth term in a sequence when you are given the function just plug in that value for n and solve. For example the 5th term of f(n)=2n-3 is f(5)=2(5)-3 =7 so the 5th term is 7.
Arithmetic and Geometric Sequences
An Arithmetic Sequence is a sequence where the difference between two consecutive terms is constant. The difference is called the common difference. For example 2, 4, 6, 8, 10, . . . The difference between two consecutive terms is always 2. So the common difference is 2.
A Geometric Sequence is a sequence where the ratio between two consecutive terms is constant. This ratio is called the common ratio. For example 3, 9, 27, 81. . . This time each term is 3 times the number before it. So the common ratio is 3.
There are two different ways to write a function for a sequence. The Explicit function is what we have been doing so far such as f(n) = 3^n. The explicit function is easier to use when you are trying to find a specific term like the 13th term but it is harder to write the function.
The recursive formula for a sequence is a function whose variable is the preceeding term.You have to give the first term of the sequence as part of the recursive formula. An example of a recursive formula for 3, 9, 27, 81 . . . is an=3(an-1) with a1=3 wherea1=3 means the first term is 1, an is the nth term and an-1 is the term right before the nth term. The recursive formula is easier to write than the explicit function but it is harder to find a specific term becuase you have to start at the first term and work your way up to that term.
Features of Functions:
Increasing or Decreasing: As you look at the graph if the y-vales are going up as you go left to right(or it has a positive slope) it is increasing. If the y values are going down from left to right (or it has a negative slope) it is decreasing. On a Chart make sure the x's are increasing as you go down the chart. If they are not, reorder the chart so that the x values are increasing as you go down the chart. Then look at the y's if they are increasing as you go down the chart the function is increasing but if they are decreasing then the function is decreasing.
X and Y Intercept: On a graph the x intercept is where the function crosses the x axis and it is written as a point (x, 0) where x is the x intercept. On a graph the y intercept is where the function crosses the y asix and it is written as a point (0, y) where y is the y intercept. On a chart the x intercept is where y is equal to 0 and the y intercept is where x is equalt to 0. There will not always be an x and a y intercept.
End Behavior: End behavior is what the y-values are doing at the ends of the function. It is two numbers, one for each end seperated either by the word "to" or by a comma. Do they go off into infinity? Does it level off at a certain number? End behavior is easy to tell from a graph by just looking at it. Assume the pattern that is shown in the graph continues. If the y values continue to get larger and larger without bounds then the end behavior is positive infinity. If the y values continue to get smaller and smaller in the negative values then the end behavior is negative infinity. If the y values level off at a certain number then the end behavior is that number. When looking at a chart look at the highest and lowest x values and look at the pattern in the y values. These will help you figure out the end behavior. Examples: "negative infinty to infinty", "0 to infinity", or "negative infinity, 4".
Positive and Negative: A graph is positive when the y values are above 0 or where the graph is above the x axis. A graph is negative when the y values are below 0 or where the graph is below the x axis. The x intercepts are the bounds between the positive and negative portions of a function. We describe where the y values are positive and negative using intervals that are the x values. So a graph could be positive on the interval -2 to 5, this would mean that the y values are positive when the x values are between -2 and 5.
Rate of Change:
Rate of Change is the slope. The formula for slope is rise over run or (Y2-Y1)/(X2-X1) where the two points are (X1, Y1) and (X2, Y2). On a graph find two points that have integer coordinates and then find the slope using the formula. On a table, pick two rows (each row is a point) and find the slope using the formula. For linear functions the Rate of Change will be constant or it will not change when you pick different points.
An average rate of change is used when the fuction is not linear. The average rate of change is the slope between two points on the function but will not be the slope at every point along that function. It is just an average of all of the slopes between those two points. To find the average rate of change use the slope formula and the specified points.
Increasing and Decreasing. An increasing rate of change is a positive slope. A decreasing rate of change is a decreasing slope. A rate of change that is increasing at an increasing rate is then a positive slope that is getting steeper or becoming a larger positive number. A rate of change that is increasing at a decreasing rate is a positive slope that is getting less steep or becoming a smaller positive number. A rate of change that is decreasing at an increasing rate is a negative slope that is getting steeper or the negative number is becoming smaller. (Remember -5 is smaller or less than -3). A rate of change that is decreasing at a decreasing rate is a negative slope that is getting less steep or the negative number is becoming larger.
Graphing Given the Function:
Linear Functions: The easiest way to graph a linear function is to put it into slope intercept form or y=mx+b where m is the slope and b is the y intercept. To find the x intercept you can either find it on the graph or substitute 0 in for y in the equationand solve for x.
Exponential Functions: Exponential functions have the general form f(x) = bx. If b is greater than 1, the graph increases. When b is between 0 and 1 then the graph decreases. If b is negative then it is reflected across the x axis. Multiplying bx by a number shifts the graph left or right. Adding a number to bx shifts the graph up or down.
Exponential graphs have asymptotes. An asymptote is a line that the graph approaches but never intersects. With the line f(x)=bx the asymptote is the horizontal line at y=0. When the graph is shifted up or down such as g(x) = bx+ a then the asymptote is at the horizontal line x = a.
Use the online graphing calculator (on my links page) to help you explore the graphs of exponential functions.
Look at the two functions and then compare and contrast them.
Rate of Change. Linear equations have a constant slope and you should be able to tell me what that slope is. Exponential functions rate of change is not constant so tell me if it is increasing at an increasing rate, increasing at a decreasing rate, decreasing at an increasing rate or decreasing at a decreasing rate. Compare the slopes. Which one is greater? Are they the same?
Intercepts. Look at the x and y intercepts. Are they the same? Are they both negative? Do they have both intercepts?
Greater or Less than. Look at the y values for the two functions. Tell me on what interval (remember intervals are x values) one of the function's y values is greater than the other function's y values.
Connecting it All
TYPO! On problem 6 change the equation from C(x) =3x+1 to C(x)=3x-3
There is not a lot I can say about this topic. Refer to the notes above and put all of the pieces together to solve the problems. Good Luck!
Writing Linear Systems and Exponential Functions from a Context
Steps to writing system of linear equations from a context.
- Define the 2 variables. (99% of the time these will be categories-like x=number of children or y=number of hotdogs- not numbers-like x=20 or y=5)
Look for the two different situations/topics/ideas that use one or both variables.
- Highlight/circle one situation.
- What information contributes to that fact/situation
Write an equation for that situation.
- How can I use my variable(s) to represent that situation/fact
- Repeat 2 and 3 for the other situation/topic/idea.
Example: The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?
Variables: c=number of children who came to the fair. a=number of adults who came to the fair
Situation: 2,200 people entered the fair. This is the total number of people at the fair. So this includes both children and adults so the equation would be 2,200=a+c
Situation: $5,050 is collected. This is the amount of money that is collected from both adult tickets and child tickets. Since this is talking about money I need to have the price of the tickets in my equation. So 5,050=(4)a+(1.50)c since an adult ticket costs $4 and a child's ticket costs $1.50.
Writing an exponential function from context
General form of an exponential function Y= (a)b^x
Is it a growth or decay problem
- Growth b >1
- Decay 0<b<1
Find the growth or decay factor
- Growth add the percent-as a decimal to 1.
- Decay subtract the percent-as a decimal from 1.
- If it is doubling, tripling, etc then b is equal to 2, 3, etc
- a is the initial amount.
Example: The population of the popular town of Smithville in 2003 was estimated to be 35,000 people with an annual rate of increase (growth) of about 2.4%. Find the population in 2007.
It is a growth problem so the b will be greater than one. To find what b is, I take the rate of growth 2.4% and turn it into a decimal 0.024. Then I add that decimal to 1 to get 1.024 which is equal to b. The intial amount of people is 35,000 so that is a. The equation is then y=(35,000)1.024^x.
Combining functions is simplying adding, subtracting, multiplying, or dividing the two functions.
For example if f(x)= x+3 and g(x)=2x-5 then
(f+g)(x) is simply f(x) + g(x) which is (x+3) + (2x-5) = 3x-2.
(f-g)(x) is f(x) - g(x) which is (x+3) - (2x - 5) = -x + 8 remember to distribute the subtraction to both the 2x and the -5 which is why you get 3 - (-5) = 3+5 = 8.
(f*g)(x) is f(x) * g(x) which is (x+3)(2x-5). remember to foil or use the distribuative property to get 2x^2+x-15.
(f/g)(x) is f(x) / g(x) which is (x+3)/(2x-5) which can't be simplified so that is just your answer.
If you have (f+g)(4) that just means to add the two functions and plug 4 in for x. So you can add the functions first to get 3x-2 and then plug in 4 for x. 3(4)-2= 10. Or or you can plug in 4 for x first and then add the functions. (4+3) + (2(4)-5)= 7 + 3 = 10.
Composition of Functions
When you compose functions you are taking the output of one function and using that as the input for the next function. The notation is f(g(x)) which is said "f of g of x". It means take the function g(x) and plug it in for x in the f(x) function. For example. f(x)=2x+5 and g(x)=4x-1. So f(g(x)) is 2(4x-1)+5=8x-2+5=8x+3. g(f(x)) means take f(x) and plug it into x in the g(x) equation. So for the same example f(x)=2x+5 and g(x)=4x-1. g(f(x))=4(2x+5)-1=8x+20-1=8x+19.
If you want to find "f of g of some number" such as f(g(3)) there are two ways you can do it. You can first find f(g(x)) which for f(x)=2x+5 and g(x)=4x-1 is f(g(x))=8x+3 and then plug in 3 for x. Then f(g(3))=8(3)+3=24+3=27. Or you can do it in pieces. First find g(3) which is 4(3)-1=11. Then take that answer or 11 and plug it into f(x) to get f(11)=2(11)+5=22+5=27. So either way you do it you get 27 as your answer.
Function transformations are different manipulations of the original function. The rate of change, y intercepts, x, intercepts, and asymptotes might change depending on the transformations. There are four different transformations that we are looking at.
Let f(x)=-3x+2 and k=5
f(x)+k means take the original function f(x) and add the variable k to the end of the function. f(x)+k=(-3x+2)+5=-3x+7. F(x)+k changes the y intercept of linear functions and the asymptote and y intercept of exponential functions.
f(x+k) means add k to x inside of the function. f(x+k)=-3(x+5)+2=-3x-15+2=-3x-13. f(x+k) also just changes the y intercept of linear functions but changes only the y intercpet of exponential functions.
f(x)*k means take the original function and multiply by k at the end. f(x)*k=(-3x+2)5=-15x+10. f(x)*k changes the slope and y intercept on linear functions and the slope and y intercept of exponential functions.
f(x*k) means multiply x by k inside of the function. f(x*k)=-3(5x)+2=-15x+2. f(x*k) changes the slope on linear functions and the slope on exponential functions.