Hi. I m a three. I m always a three. I never ever change. That s why I m a constant.

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Sophia Griffith
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1 Lesson 1-1: 1 1: Evaluating Expressions Hi. I m a three. I m always a three. I never ever change. That s why I m a constant. 3 Real life changes, though. So to talk about real life, math needs things that change. We need variables. Hi. X Any time you have variables in math, you re dealing with algebra. A collection of constants, variables, and operators (addition, multiplication, whatever) is an algebraic expression. Ex. 1: Gimme a bunch of algebraic expressions. Switching gears Let s suppose you have to multiply a number by itself many times. Rather than writing out It s time to get lazy. Let s turn that into 10 We ll call that the base and the 10 the exponent.. Together, it s a power or exponential expression.

2 Ex. : Express each of these in words and as a product. a. 6 3 b. p c. 1 d. x 0 Exponential expressions are useful all over, but it s easy to see it in geometry. Ex.3: a. Find the area of this square: 3 in. b. Find the volume of this cube: cm

3 Lesson 1-: 1 Order of Operations Ex. 1: Solve: Mathematicians have come to generally agree on the order you do your math in, and it ain t always left to right. Here s your order in two different styles: Parenthesis Exponents Multiplication Division Addition Subtraction Brackets Other stuff Division Multiplication Addition Subtraction Ex. : Solve: [ ] 1 a. 10 ( + 3) + b. 3 3 (1 + 3)

4 Lesson 1-6: 1 Utilizing Functional Nomenclature (aka. Using math words) Suppose you re trying to solve an algebraic expression from a word problem. Let s use something simple like The yacht floats 37 above the water. How high above the lake bed is the yacht? Equation: Can you use any number for the variable? Probably not. The water level can t be below the bottom, and it can t get too high for various reasons. There s a limit to the numbers you can stick in for that variable, and therefore there s a limit to the numbers you ll get out of the equation. Mathematicians felt a burning desire to name these: Domain: what you can stick into an equation, table, function,, etc. Range: : what you can get back out Ex. 1: Come up with a reasonable domain and range for the yacht equation. Domain: Range: Ex. : What s the domain and range of this table? t Input Output 3 3 1

5 A quick example of a mapping diagram: Ex. 3: : Identify the domain and range of that thing In an equation, your domain (input) is determined by the independent variable.. The range (output) is determined by the dependent variable (dependent because it depends on the input). Mathematicians are especially concerned with inputs that only give one output (or any instance of the independent variable will only produce a single result in the dependent variable ). These are called functions. That s a lot of mathspeak. Probably a good time for some examples. Ex. : In the yacht equation, what was the independent variable? What was the dependent variable? Ex. 5: : Is the mapping diagram above a function? Ex. 6: : Give examples of a table and a mapping diagram that aren t functions.

6 Lesson 1-7: 1 Representing Functions as Graphs Simple functions are pretty easy to understand. For instance: y = x +1 When x goes up More complicated functions can be hard to get a feel for just by looking at the equation. 3 y = x x x + x + 1 When x goes up or down or??????? You can often get a better feel for what a function does by looking at it visually instead of algebraically. Turn it into a graph. (For now, we ll stick with graphing simpler functions, but you ll learn to deal with more complicated functions as you take more classes.) To make a graph from a function by plotting: Use the horizontal (left-right) axis for x and the vertical (up- down) axis for y Pick an (x, y) pair using the equation (you can also make a table of many points) Plot each point using the axes Repeat until a trend shows up or you re told to stop Ex. 1: Graph the function y = 1 x 3 using the domain 0, 3, 6, and 9 Try to pick a reasonable scale for your axes. What happens if you don t?

7 Sometimes you need to work backwards from graph to equation. In that case, 1. Make a table of a few points. See if you can figure out a relationship that works 3. Write a function to represent that relationship Ex. : Write an equation for the function represented by this graph: But wait! There s more! How do you know if something s s a function or not? (Reach alllllllllll the way back to the previous lesson ) Ex. 3: Is this a function? Input a. Output b. X Y There s a pretty simple way to find out if something s a function when you re looking at a graph: Does it pass the vertical line test? That is, does any vertical line only touch the graph once?

8 Ex. : Is this a function? a. b

Project 2: How Parentheses and the Order of Operations Impose Structure on Expressions

MAT 51 Wladis Project 2: How Parentheses and the Order of Operations Impose Structure on Expressions Parentheses show us how things should be grouped together. The sole purpose of parentheses in algebraic