Do you ever understand something but not understand it at the same time? That's where I'm at right now.
That's a lot to chew on at one time....when do you need this by ??
Well..I'll see what I can do....but...I can't promise that I can get to each one!!!
Th quadratic is y = x^2 [a parabola ]
Here's the graph : https://www.desmos.com/calculator/3gdzi6zvsn
End Behavior : Up at both ends
Average Rate of Change [ I don't see how your teacher expects you to know this ... you need Calculus]....there is no "average rate of change"....the rate of change [slope of the graph ] at any value of x = 2x
Positve/Negative ..I assume this refers to the range ??...The graph is never negative for y...in other words, the range is [0, infinity )
x intercept and y intercept = (0,0)
How used in the real world.....If you have a sattelite dish....it is a parabola....the satellite beams the signal to the dish...it is reflected back to the focus of the parabola and transmitted to your inside receiver
The graph is of y = l x l .....https://www.desmos.com/calculator/wn5b3geed5
End behavior : Up at both ends
Average rate of change : on (-inf, 0) the average rate of change = -1....on (0, inf), the average rate of change is 1
Positiive / Negative - the graph is never negative...thae range is [ 0, inf )
X, y intercept = (0,0)
How used in the real world : Can't think of any....but I'm sure there are some !!!
See the graph of y = https://www.desmos.com/calculator/cooqsnkeug
End behavior.....this graph always rises from left to right on its domain.....so..."Up" on the right
Average Rate of Change : THere is no "average rate of change"...again....this requires Calculus...the "average rate of change " is the slope at any point on the graph [except x = 0 ] and is given by (1/2) x ^(-1/2)...at x = 0 th slope is undefined
Positive/Negative : The graph has a range of [0, infinity) it is never negative
x / y intercept = (0,0)
Again....I can't name any real world applications off the top of my head....(sorry !! )
See the graph of y = log (x) here : https://www.desmos.com/calculator/nxcaogf186
End Behavior : Near x = 0, the graph turns downward.....as the graph increases from left to right, its end behavior is "up" on the right
Avg Rate of Change....there is no "Average rate of change"...again Calculus will show that the slope at any point x is :
1 / [x ln 10 ]
Positive/ Negative : the graph is negative [ below the x axis} on (0, 1)....the graph is above the x axis on (1, inf)
x, y intercept : this graph never intercepts y, because we can't take the log log of x = 0.....the x intercpt is at x = 1
Real World : logs are used in the calculation of magnitudes of earthquakes
See the graph of y = 2^x : https://www.desmos.com/calculator/btsykhud9v
End Behavior...at the left end, the graph approaches y = 0 ...at the right end it approaches infinity = "up"
Avg Rate of Change ;: Calculius again....there is no average rate of change....th slope at any x value is 2^x * ln (2)
Positve/ Negative : The graph is positive for all x...the range is (0, inf )
x , y intercepts - no x intercept...y intercept is (0,1).... [which is true for all graphs of the form y = a^x]
Real world : Used to model growth rates in populations, etc.
Last One [ you owe me BIG TIME...LOL!!! ]
These might be anything but here is the graph of y = (x + 1) /( x - 1) : https://www.desmos.com/calculator/watidrudhk
[ This is actually the graph of a "rotated" hyperbola ]
End Behavior : Far left end : UP ...as we approach x = 1 from the left : DOWN....as we approach x = 1 from the right : UP.....as we approach infinity : DOWN
Avg Rate of Change...Calculus, again...there is no "average rate of change"...the slope at any point on the graph is :
-2/ (x -1)^2
Positive/ Negative : The graph is above the x axis on (-inf, -1) U ( 1, inf)..it is below the x axis on ( -1, 1)
x, y intercepts : intercepts x axis at (-1,0) and the y axis at (0, -1)
Real World : If you find any, let me know !!! LOL!!!