The graph shows how the cost of hakims birthday party depens on the number of guests

9514 1404 393

  $1.50 per guest

Step-by-step explanation:

The rate of change is the slope of the line, the "rise" divided by the "run".

For computing the slope, it is convenient to use points where the line crosses grid intersections. We note that both ends of the line shown are at grid intersections, so we'll use those points.

  m = rise/run = (10 -1)/(6 -0) = 9/6 = 3/2 = 1.50

The rate of change is 1.50 dollars per guest.

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.