hectictar

The average rate of change of  f  over the interval  2.6 ≤ x ≤ 5.2  is the slope of the blue dotted line on the above graph.

The blue dotted line passes through the purple point,  (2.6, f(2.6))   , and the green point,  (5.2, f(5.2))

Do you know how to find the slope of the blue dotted line? (Remember "rise over run" )

Here's one way to solve this:

The equation of this parabola in vertex form is

y  =  a(x - 2)²  -  4     , where  a  is some constant that "stretches" or "shrinks" the parabola.

To find  a  we can use the fact that the parabola passes through the point  (4, 12)

So we know the following must be true:

12  =  a(4 - 2)²  -  4

                                Simplify the right side of the equation....

12  =  4a  -  4

                                Add  4  to both sides of the equation.

16  =  4a

                                Divide both sides of the equation by  4

4  =  a

So the equation of this parabola is

y  =  4(x - 2)² - 4

To find the zeros, we set  y  equal to  0  and solve for  x

0  =  4(x - 2)² - 4

                              Add  4  to both sides of the equation.

4  =  4(x - 2)²

                              Divide both sides of the equation by  4

1  =  (x - 2)²

                              Take the ± square root of both sides.

±1  =  x - 2

                              Split into two equations and solve for  x ...

x - 2  =  1       or       x - 2  =  -1

  x  =  3                         x  =  1

So the zeros of the quadratic are at   x = 3   and   x = 1

3 - 1  =  2

C  =  G - 3F

when

F  =  2x²+ 6x - 5

and  G = 3x² + 4

So we just have to substitute   2x²+ 6x - 5   in for   F   and   3x² + 4   in for   G

C  =  G - 3F

C  =  (3x² + 4) - 3(2x²+ 6x - 5)

C  =  (3x² + 4) - 3(2x²+ 6x - 5)

The next steps are to simplify expression.

First distribute the negative three to the terms in the parenthesees.

Second combine like terms.

Perfect!! 

Very close!! But be careful with your negative signs!

g(-2)  =  -4(-2)² - 3(-2) + 2

                                              Negative two squared is equal to positive four.

g(-2)  =  -4(4) - 3(-2) + 2

                                              Negative four times positive four equals negative sixteen.

g(-2)  =  -16  - 3(-2) + 2

                                              Negative three times negative two equals positive six.

g(-2)  =  -16  +  6  +  2

Now just calculate negative sixteen plus six plus two!

g(x)  =  -4x² - 3x + 2

                                              Replace every  x  with  -2

g(-2)  =  -4(-2)² - 3(-2) + 2

Now what do you get if you calculate the right side of that equation?

My guess:

Total number of arrangements of  0, 1, 6,  and  7   =   4!   =   24

For each of these, there are 2 possible choices for the first three digits.

So the total number of possibilities  =  24 * 2   =   48

Only one of these 48 possibilities is correct, so the probability that Fatima chooses the correct one is

1 / 48

----------(below is my reassurance that there are actually 24 ways to arrange the last 4 digits!)----------

0167

0176

0617

0671

0716

0761

1067

1076

1607

1670

1706

1760

6017

6071

6107

6170

6701

6710

7016

7061

7106

7160

7601

7610

We want the discriminant to be negative, so we want

b² - 4(1)(9)  <  0

b² - 36  <  0

b²  <  36

This is true when...

| b |  <  6

Which is true when...

b < 6     and     b > -6

So  b  is any number in the interval  (-6, 6)

(The endpoints are not included because that makes the discriminant equal to zero, which gives only one real root to the original polynonmial.)

g(x)  =  -2x² + 3x

                                        Replace every  x  with  -3

g(-3)  =  -2(-3)² + 3(-3)

                                        Evaluate the right side of the equation.

g(-3)  =  -2(9) - 9

Can you finish it from here?