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)
