Algebra & Algebra Honors

Interpreting Graphs of Functions 1-8

There are several key features of different functions that help you identify what type of function it is and also interpret how it’s going to act.
 LINEAR OR NONLINEAR:

If a graph has a curve, it’s nonlinear. If it’s a straight line, it’s linear.
You can see this easily when it’s graphed.
On the graphing calculator, you’ll discover that if the x power is 1, it’s a line (linear)
When we change the x power to 2 or 3 or higher, it’s nonlinear.

INTERCEPTS:

These are points where the graph intersects the x or y axis.
x-intercept: where the graph intersects the x axis…the coordinate would be of the form (x, 0)
y-intercept: where the graph intersects the y axis…the coordinate would be of the form (0, y)

If the graph goes through the ORIGIN, both intercepts would be (0, 0)
A horizontal line would not have an x-intercept UNLESS the line is the x axis (the y value would always be 0 or y = 0)
A vertical line would not have an y-intercept UNLESS the line is the y axis (the x value would always be 0 or x = 0)

IS IT POSSIBLE FOR A GRAPH TO HAVE MORE THAN ONE X OR Y INTERCEPT???
If it’s a line (linear), NO. A line can’t come back around again.
However, if a graph has a curve (nonlinear), YES it can…it can intersect say the x axis and then curve around and intersect the x axis again.

MOVING THE Y-INTERCEPTS UP OR DOWN:Adding a POSITIVE constant at the end of a function moves the graph UP and adding a NEGATIVE constant moves it DOWN.
y = x goes through the origin           y = x + 2 moves it up 2            y = x – 3 moves it down 3

 SLOPE:
When the coefficient of x is POSITIVE, it looks like you’re going up the mountain.
When the coefficient of x is +1, the slope going up is a 45 degree angle.
As the coefficient of x gets greater than 1, the steepness of the line INCREASES.
As the coefficient goes into the range between 0 and 1 (a fraction or decimal), the slope starts to level out.
When the coefficient is negative the line switches direction and looks like you’re going down the mountain.

 SYMMETRY:Just as you learned in geometry, line symmetry means that one half of a graph looks like
the other half along some vertical line.
We’ll see that y = x² is symmetrical along the y axis.
If we move the graph to the right so it’s all in the first quadrant and look at it as the trajectory of a ball, the symmetry could be interpreted as it took the same amount of time for the ball to rise up in the air as it did to come down.

POSITIVE AND NEGATIVE PARTS OF A GRAPH: This is pretty obvious!
A function is positive where the graph is ABOVE the x axis…the RANGE is positive above the x axis.
A function is negative where the graph is BELOW the x axis…the RANGE is negative below the x axis.

INCREASING AND DECREASING PARTS OF A GRAPH:When the graph is going UP, the function is INCREASING.
When the graph is going DOWN, the function is DECREASING.
REMEMBER WE’RE LOOKING AT THE GRAPH FROM LEFT TO RIGHT!

EXTREMA:Extrema comes from the word extreme so we’re talking about extreme values of a function…either high range values or low range values (y values)
There are two kinds of extrema: minimums and maximums
A minimum means that there are no other y values (range values) lower anywhere in the function
A maximum means that there are no other y values (range values) higher anywhere in the function

A RELATIVE minimum means there are no other y values lower NEARBY (but there may be lower points in another region of the function)
A RELATIVE maximum means there are no other y values higher NEARBY (but there may be higher points in another region of the function)

 END BEHAVIOR:

Every graph has an “end” on both sides of the domain values (x values)
As x gets smaller towards negative infinity (meaning you’re going to the left on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

As x gets larger towards positive infinity (meaning you’re going to the right on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

Generally, we summarize end behavior by comparing what x (the domain) is doing to what y (the range or function value) is doing at the same time:

 As x decreases—>y also decreases OR  y increases

As x increasesà y also increases OR y decreases

DOMAIN AND RANGE ON A GRAPH:You already know that the x values are the domain and y values are the range.
On a graph, we look at all the possible x values to determine if the domain is all real numbers or if it’s limited in some way.
We do the same thing with the range.

For example: f(x) = x²
This is a U shaped graph that only goes up from the origin so the range is limited to y ≥ 0
The domain would be all real numbers because you can square any number and, looking at the graph, you can see that eventually the graph will go to both negative and positive infinity to the left and to the right.

REAL WORLD INTERPRETATIONS OF GRAPHS:

Sales of a company:
By looking at a graph of sales over time, you can analyze how the company is doing.
The increasing parts of the graph mean that the company is growing while the reverse is also true.
If you see a flat part of the graph, that part would show the company is staying the same.

Between an increasing and decreasing part of sales would be a relative max to sales…meaning for some reason the company is in decline.

Between a decreasing and increasing part of sales would be a relative min to sales…meaning for some reason the company is doing well again.

The end behavior over time TO THE RIGHT would predict the success of the company in the future. (to the left would be the actual history of sales)

Math 8

CHAPTER 2-2: SOLVE TWO-STEP EQUATIONS

1. Use the ADDITION/SUBTRACTION PROPERTIES OF EQUALITY

(get rid of addition or subtraction)

2. Use the MULTIPLICATION/DIVISION PROPERTIES OF EQUALITY

(get rid of multiplication/division)

If both the constant is a fraction and the coefficient is also a fraction, sometimes it’s better to get rid of the denominators by multiply by the LCM of all the denominators first:

When there is addition or subtraction in the numerator and a constant (number) in the denominator, multiply by the denominator first!

Then you’ll have a simple one-step equation after that ;)

EXAMPLE:

Math 8

CHAPTER 2-1: Equations in One Variable

REVIEW OF SIMPLE EQUATIONS! One and Two Step!

GOAL? Determine the value of the variable

HOW? Isolate the variable (get it alone on one side of equation)

WHAT DO I DO? Use inverse (opposite) operations to "get rid" of everything on the side with the variable

WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED?

Always focus on the variable(s) first!!!!!!!

We'll be meeting our old BFFs from 7th grade!

EQUATION BALANCING PROPERTIES OF EQUALITY:

Whatever YOU DO TO BALANCE an equation,

that operation is the property of equality that was used.

If you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally.

If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally.

If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3.

If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3.

SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY

CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?

Since we say we never subtract and we really never divide, it's those 2.

GOING BACK TO OUR PREVIOUS EXAMPLES:

If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.

If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY.

(We always multiply by the MULTIPLICATIVE INVERSE).

Special type of one-step equation are those where the VARIABLE IS NEGATIVE.

Remember: You’re solving for the POSITIVE VARIABLE.

There are a couple of ways to do this.

DID YOU KNOW THAT YOU CAN MOVE A NEGATIVE SIGN

IN 3 DIFFERENT PLACES ON ANY FRACTION????

So if you see a negative sign on a variable in a fraction, you can move the negative sign!

We use the MULTIPLICATIVE INVERSE PROPERTY to balance the equation and ISOLATE the variable on one side.

The multiplicative inverse of a number is its reciprocal.

The multiplicative inverse of ¾ is 4/3.

If the fraction is NEGATIVE, so is its multiplicative inverse:

Example: If you have -3 ½ , first make it improper: -7/2, and then its multiplicative inverse (reciprocal) is -2/7.

MULTIPLY BY THE RECIPROCAL ON BOTH SIDES TO SOLVE A ONE-STEP EQUATION WHERE THE COEFFICIENT IS A FRACTION.

FOR DECIMALS, SIMPLY DIVIDE EACH SIDE BY THE DECIMAL AND DO THE DIVISION USING YOUR UNDERSTANDING OF LONG DIVISION WITH DECIMALS.

I’ll also show you how to get rid of the decimals as a different approach.

If you’d like to try this approach for decimals:

Multiply both sides of the equation by a POWER OF 10 big enough to get rid of the all the decimals (make them all integers).

Example: 3.45x = .005

You would need to multiply both sides by 1000 so that both sides would no longer have decimals:

1000(3.45x) = 1000(.005)

3450x = 5

Now you would divide by 3450 on both sides.

Note: You’ll still get a decimal answer, but while you’re solving you don’t have decimals.

This works really well when all the terms have the same place value!

If the coefficient is a fraction, you’ll get the answer in ONE-STEP if you use the MULTIPLICATIVE INVERSE PROPERTY and multiply both sides equally by the RECIPROCAL of the coefficient.

Example:

¾ x = 15

(4/3)(¾ x) = (15)(4/3)

x = 20

MAKE SURE YOU ALWAYS CROSS CANCEL if possible!!!

Algebra & Algebra Honors

Function 1-7

Function: a relation (set of ordered pairs) where there is EXACTLY ONE output for each input. Each element of the domain has EXACTLY ONE element in the range. THE X VALUES NEVER REPEAT!

Vertical Line Test: If the relation is represented with a GRAPH, this test is the easiest way to see if an x value repeats. Draw vertical lines up and down continuously on the graph and see if a line intersects with (hits) more than one point. If it does, it’s a relation, but not a function. If it doesn’t, it’s a function.

***Discrete function: a function where the ordered pairs are not connected (For example, you can’t purchase a part of a candy bar at 7-11)

***Continuous function: a function where the ordered pairs are connected in a smooth curve (For example, if you’re driving in a car, the distance ever increases continuously)

Function notation: f(x): If a relation is a function, you can write the equation using y as a variable as the function value OR you can use f(x) as the function value. You read this as “the function of x” or the function value for the given x value.  Note that “f” is NOT A VARIABLE…it’s an abbreviation for the word FUNCTION…so don’t ever divide by f

Example: y = 2x + 3 OR f(x) = 2x + 3 represent the same function.

To find f(2) in the above function, simply plug in 2 for x and evaluate: f(2) = 2(2) + 3 = 7 so f(2) = 7

WHAT’S BETTER ABOUT f(2)=7 vs y=7 although they mean the same thing?  In function notation, you know both the domain value and the range value!    Using other letters with function notation:

Another good thing about function notation is that you can use specific letters that show the relationship between two variables. For example, the cost of what you spend depends on how much you buy. Say you’re only buying pizzas for a big party. Let c represent the cost of the pizza and p represent the number of pizzas you purchase. The function notation of c(p) would be expressed in words as “the cost of the pizza”.  Notice: the variable inside the ( ) is the input/independent variable/domain and the outside variable is the output/dependent variable/range. … What you spend depends on the number of pizzas you order!

 You can multiply or divide functions. The way you express this is to simple show the operations on the f(x):   2[f(x)] means to double the function  Example: If f(x) = 2x + 3 then   2[f(x)] = 2(2x + 3) = 4x + 6

 LINEar function: A set of ordered pairs that draws a straight line (that’s not vertical)

NonLINEar function: A set of ordered pairs that does NOT draw a straight line