PLEASE HELP MEEEEEEEEEEEEEEEEEEEEEEEEE!!!!!!!!!!!!!!!!!!!!!
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Answer:
- lower right (see attachment 1)
- 39
- linear/exponential (see attachment 2)
Step-by-step explanation:
<h3>1.</h3>An exponential function is one that increases or decreases at a rate that is proportional to its value. That is, if it is increasing, it does so at an increasing rate. If it is decreasing, it does so at a decreasing rate.
The "base" of the function is the number in parentheses when the function is written as it is here. The multiplier is the number to the left of these parentheses, and the exponent is the small number or expression to the upper right of the parentheses.
If the "base" is less than 1, then the function will be decreasing. If there is no added constant, it will decrease toward zero. If there is an added constant (as here, -1), then the function will decrease toward that value. The only graph showing a function decreasing toward -1 is the one at lower right (attachment 1).
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<em>Note about the multiplier</em>
If the multiplier is negative (not the case here), then the graph will be reflected across the x-axis and the descriptors "increasing" and "decreasing" get reversed. A graph that would go down to the right with a positive multiplier goes up to the right with a negative multiplier.
What you can <em>always</em> say is that when the base is less than 1, the graph is steep on the left and flat on the right. For a base greater than 1, the reverse is true.
<h3>2.</h3>The average rate of change of a function between x1 and x2 is given by the formula ...
... average rate of change = (f(x2) -f(x1))/(x2 -x1)
Here, you have x2 = 10, so f(x2) = 3^(10/2) = 3^5 = 243; and x1 = 4, so f(x1) = 3^(4/2) = 3^2 = 9.
Then the average rate of change is ...
... average rate of change = (243 -9)/(10 -4) = 234/6 = 39 . . . flowers/day
<h3>3.</h3>When faced with determining the sort of function you have, first of all check the given x-values. If they are evenly spaced, as here, then you are in luck.
Next look at the differences of y-values. These are the "first differences". For the first table, they are ...
- 7 - 5.5 = 1.5
- 8.5 - 7 = 1.5 . . . . same as the other one
When the first differences are constant, as here, the function is linear.
In the second table, the first differences are decidedly <u>not</u> constant, so the function is not linear. Another test you can do is to see if the ratios of y-values are a constant. Here, those ratios are ...
- 8/1 = 8
- 64/8 = 8
These ratios are constant, so the function is exponential.
Your problem statement makes no reference to p or q, so we can only guess that "p/q" refers to the ratio of the lowest- and highest-degree coefficients:
... -6/1
Then the factors of p/q are the factors of 6, with either sign.
Factors of p/q: ±1, ±2, ±3.
There are a number of ways the factors can be tested. To test 1, simply add the coefficients. (The sum is -8.) To test -1, subtract the sum of odd-degree coefficients from the sum of even-degree coefficients. (The difference is 0.)
Knowing that -1 is a root of the function, you can use synthetic division to find the quadratic factor. (See the first attachment for the work.) Then you have ...
... f(x) = (x +1)(x² +x -6)
You can continue to test the factors of p/q, or you can factor the quadratic to get ...
... f(x) = (x +1)(x -2)(x +3)
Another way to test the factors is to program your graphing calculator to evaluate the function f(x) for those factors. The result from my TI-84 is shown in the second attachment. That shows f(x) = 0 for x = -3, -1, and +2.
Test the factors: f({-3, -2, -1, 1, 2, 3}) = {0, 4, 0, -8, 0, 24}
This tells you the x-intercepts.
X-intercepts: -3, -1, 2
Of course, the y-intercept is just the constant in the function—the value of the function when x=0.
Y-intercept: -6
As with any odd-degree polynomial with a positive leading coefficient, the overall behavior looks like "/". That is ...
End behaviors: f(x) goes to -∞ as x goes to -∞; f(x) goes to +∞ as x goes to +∞.
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A graph of f(x) is shown in the third attachment. (Graphing calculators make it easy to use graphing to find x-intercepts.)
