hectictar

y   =   2x² + kx + 6

                                        We are given that   y = -x + 4  , so we can substitute  -x + 4  in for  y .

-x + 4  =  2x² + kx + 6

                                        Add  x  to both sides of the equation.

4   =   2x² + kx + x + 6

4   =   2x² + (k + 1)x + 6

                                        Subtract  4  from both sides of the equation.

0   =   2x² + (k + 1)x + 2

This equation will have one solution when...

(k + 1)² - 4(2)(2)   =   0

(k + 1)² - 16   =   0

(k + 1)²   =   16

k + 1   =   ± 4

k   =   ± 4 - 1

k   =   4 - 1   =   3          or          k   =   -4 - 1   =   -5

The functions are inverses of each other.

The graphs of functions are symmetric to each other over the line  y = x .

First we need to find the slope of the line...

y - 5  =  6x - 10

                            Add  5  to both sides of the equation.

y  =  6x - 5           Now that this is in slope-intercept form, we can see that its slope is  6 .

The slope of a line that is parallel to this will also be  6 .

Using the point  (4, 10)  and a slope of  6  , the equation of our line in point-slope form is...

y - 10   =   6(x - 4)

                               Distribute the  6  to both terms in parenthesees.

y - 10   =   6x - 24

                               Add  10  to both sides of the equation.

y   =   6x - 14          This is the equation in slope-intercept form.

lynx7, please don't copy an answer without giving credit.

If you find the answer on the forum already, just put a link to it...like this...

Since the question only mentions magnitude and intensity, I guessed that a "stronger" earthquake meant that it has a higher intensity.

Let  I₁  be the intensity of the earthquake with magnitude  8 .

Using the formula given in your question...

8   =   log( I₁/S )          Solve for  I₁ .

                                   Take the inverse of the log of both sides.

10⁸   =   l₁/S

                                   Multiply both sides by  S .

10⁸S   =   l₁

Let  I₂  be the intensity of the earthquake with magnitude  6 .

6   =   log( l₂/S )

10⁶S   =   l₂

The question is...    l₁   equals what times  l₂  ?

l₁  =  x * l₂

10⁸S   =   x * 10⁶S

[ 10⁸S ] / [ 10⁶S ]   =   x

10⁸ / 10⁶   =   x

100   =   x

So...

l₁  =  100  *   l₂

An earthquake with a magnitude 8 is 100 times more intense than an earthquake with magnitude 6 .

(a)

the amount of money after  0  years   =   500

the amount of money after  1  year     =   500  *  1.015

the amount of money after  2  years   =   500 * 1.015 * 1.015

the amount of money after  t  years   =   500 * 1.015^t

f(t)   =   500 * 1.015^t

(b)

We want to find the value of  t  when  f(t) = 800

800   =   500 * 1.015^t

1.6  =   1.015^t

log( 1.6 )  =   log( 1.015^t )

log( 1.6 )   =   t log( 1.015 )

log( 1.6 ) / log( 1.015 )   =   t

31.568   ≈   t

Since interest is added annualy, it would take  32  years for the account to have  $800 .

( After  31  years, it  has  $793.26  .  After 32 years, it has  $805.16 .)

If the number that is being raised to the power of  t is in the interval  (0, 1) , it is exponential decay.

If the number that is being raised to the power of  t  is greater than  1 , it is exponential growth.

For example,

y  =  3.7(0.2)^t       Here,  0.2  is being raised to the power of  t .

0.2  is in the interval  (0, 1)  so this equation represents exponential decay.

See if you can figure out the others.

And for this one...

g(x)  =  2.1(x)

Here, there is no exponent. So this equation is neither exponential growth nor exponential decay.

f(x)   =   -0.25(6)^x+2 - 1

The y-intercept is the value of the function when  x = 0 .

Plug in  0  for  x  .

f(0)   =   -0.25(6)⁰⁺² - 1

f(0)   =   -0.25(6)² - 1

f(0)   =   -0.25(36) - 1

f(0)   =   -9 - 1

f(0)   =   -10

The y-intercept is  -10

the number of bacteria after  0  hours   =   250

the number of bacteria after  1  hour   =   250 * 4

the number of bacteria after  2  hours   =   (250 * 4) * 4

the number of bacteria after  3  hours   =   ( (250 * 4) * 4 ) * 4   =   250 * 4 * 4 * 4   =   250 * 4³

the number of bacteria after  x  hours   =   250 * 4^x

y   =   250 * 4^x