CPhill

No prob...maybe they should call Me "Lightning,"  huh ??

5x^2 + 19x + 16  > 20      subtract 20 from both sides

5x^2 + 19x - 4   > 0       let's set this to 0

5x^2 + 19x - 4  = 0     factor, if we can

(5x -1 ) ( x + 4)  = 0      set each factor to 0 and solve for x and we  get

x  =1/5     and  x = -4

Note that  we have three possible intervals that make the original inequality  true

(-inf, -4)  or  ( - 4, 1/5)  or  ( 1/5, inf )

Normally, in a quadratic...if the middle interval  makes the inequality untrue, the  outside intervals will make it  true

Testing a point in the middle interval  - I'll pick  x  = 0 -  and subbing it into the original inequality  produces  16 > 20

So....the  middle interval makes the inequality untrue....and the integers in this interval  are  -3, -2, -1 and 0  [ 4 values]

(Note that we  can't include -4 since it is not part of the interval itself)

See the graph, here : https://www.desmos.com/calculator/dkmaryyfcp

We have no restriction on the domain  for the second function  because we can take the cube root of any real number

However....for the first function,  x cannot be  < 2  because we would have a negative square root which does not produce a real  value

So......we must take the most restrictive domain  for both functions....this is  [ 2,  infinity )

Here's a graph : https://www.desmos.com/calculator/ant19juary

Given that the point (4,7) is on the graph of y=f(x), there is one point that must be on the graph of 2y=3f(4x)+5. What is the sum of coordinates of that point?

2y  = 3f(4x) + 5       divide through by  2

y = (3/2) f(4x) + 5/2

The  "4x"  part  compresses the graph horizontally by a factor of 4

The "3/2"  part vertically  stretches the graph by a factor  of 3/2

The "5/2"  part  shifts the resulting point up by  5/2 units

Applting the first two  to (4,7)  we get  (4*(1/4) , 7* (3/2) ) =  ( 1, 21/2)

Applying the last one, we arrive at   ( 1, 21/2 + 5/2)  = (1, 26/2)  = ( 1,13)

The sum of the coordinates  is  1 + 13   = 14

9. Assume that $f(3) = 4$. Name a point that must be on the graph of .y = (1/2) f (x/2)

The  "x/2"  part  horizontally stretches the graph by a factor of 2   and the "1/2"  part out front vertically compresses the graph by a factor of 1/2

So....the point   ( 3, 4)  is on the original graph....and the point  ( 2*3, 1/2 * 4)  =  (6, 2)  is on  y =(1/2)f (x/2)

8. Assume that f(3) = 4. Name a point that must be on the graph of y=f(x)+4.

The point  ( 3,4)   is on  the original graph

The graph  of  y = f(x) + 4     would shift this point  upward by  4  units

So....the point  ( 3, 4 + 4)  =  (3, 8)  is on  y  =f(x) + 4

6. Let a and b be real numbers, where a < b, and let A = (a,a^2) and B = (b,b^2). The line AB (meaning the unique line that contains the point A and the point B) has x-intercept (-3/2,0) and y-intercept (0,3). Find a and b. Express your answer as the ordered pair (a,b).

We can  find the slope of the line  as  

[ 3 - 0 ] / [ 0 - -3/2] =  3 / (3/2)  =  2

And the equation of this line  is

y = 2x + 3     (1)

If   A  = (a, a^2)   and B  = (b,b^2)   then the function that would contain these points is  y  =  x^2   (2)

Setting  (1)  equal to (2), we have that

x^2  = 2x + 3

x^2 - 2x - 3  = 0     factor

(x - 3) ( x + 1)   = 0

Setting each factor to 0  and   solving for x produces  x = 3   and x  = -1

So....a  = -1  and  b  = 3  ⇒  (a,b)   = (-1,3)

Here's a graph : https://www.desmos.com/calculator/e5ys1u6aps

7. Find all points (x,y) that are 13 units away from the point (2,7) and that lie on the line x - 2y = 10. Give your answer as a list of points separated by semicolons, with the points ordered such that their $x$-coordinates are in increasing order. (So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)

Rewrite the equation of the line as   x  = 2y + 10       (1)

We can envision  that (2,7)  is the center  of a circle with a radius of 13

The equation of this circle is  (x - 2)^2  + ( y - 7)^2  = 13^2        (2)

Sub  (1)  into (2) for x  and we have

( 2y + 10 - 2)^2  + ( y - 7)^2  = 13^2   simplify

( 2y + 8)^2  + ( y - 7)^2   = 169

4y^2 + 32y + 64  + y^2 - 14y + 49  = 169

5y^2 + 18y - 56  = 0        factor as

(5y + 28) ( y - 2)  =  0

Setting each factor  to 0 and solving for y produces  y = -28/5    and  y  = 2

Sub these back into (1)  to find the associated x values

x = 2(-28/5) + 10  =  -56/5 + 50/5 =  -6/5

x = 2(2) + 10   =  14

So...the points on the line  x - 2y = 10 that are 13  units away from (2,7)  are   

(-6/5, -28/5)  and  ( 14, 2)

Here's a graph : https://www.desmos.com/calculator/37bqbov5mv

√ [ (x - 3)^2  - (x - 8)^2 ]

As small as the expression under the radical  can be  is  0

So....setting it to 0, we have

(x - 3)^2  - (x - 8)^2   =  0 

Factor as a difference  of squares

[ (x - 3) + ( x - 8)] [ (x - 3) - ( x - 8) ]  = 0

Settting the second factor to 0  and solving for  x will not produce a solution

Setting the  first factor to 0 and solving for x  produces

( x - 3) + ( x - 8)  = 0

2x - 11  = 0

x  = 11/2 

So....the smallest x value in the domain is  x  = 11/2

Here's a graph that confirms this : https://www.desmos.com/calculator/qegganaegs

g(x) = -f(x)  will "flip" the graph of f(x) over the  x axis.....it will intersect with f(x)  in two places - at the points where f(x)  intersects the  x axis...so  "a"  = 2

h(x)  = f(-x) will  "flip" the graph of f(x) across the  y axis..it wil intersect with  f(x)  in one place - at the point where f(x) intersects the y axis...so...."b" = 1

So  10a + b  =   10(2) + 1  =   20 + 1   = 21