Algebra Honors ( Periods 4 & 7)

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)

Algebra (Period 1)

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 increases à 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)