A quadratic function has these characteristics: x = 1 is the equation for the axis of symmetry. x = –1 is an x-intercept. y = –4 is the minimum value. Determine the y-intercept of this parabola.

Question

Question #1:

A quadratic function has these characteristics:

x = 1 is the equation for the axis of symmetry.

x = –1 is an x-intercept.

y = –4 is the minimum value.

Determine the y-intercept of this parabola.

Question #2:

The cost function for a telephone manufacturing plant is  and the revenue function is , where x is the number of telephones sold, in thousands.

a) Determine the profit function for the number of phones sold.

b) Determine the number of telephones sold that maximizes profit.

Answer 1

(x-3)(x+1)

parabola is y=x^2 -2x -3

x intercepts are equidistant from the vertex x coordinate = axis of symmetry, -1 and 3 are both 2 away from 1, so the x intercepts are -1 and 3

then factors are (x+1)(x-3) = x^2 -2x -3

y intercept is when x = 0, y=-3

complete the square

x^2 -2x +1 -3 -1

= (x-1)^2 -4 in vertex form, vertex = (1,-4) = minimum point

profit = R-C = -2x^2 +27x - 5x -24= -2x^2+22x-24

-2(x^2-11x+121/4) -24+726

-2(x-11/2)^2 +702 where 11/2 = 5,500 = profit maximizing phone sales, $702,000 = max profit

or use differential calculus

(R-C)' =-4x +22=0

x =22/4 = 11/2 = 5.5 thousand

Max Profit = -2(11/2)^2 +22(11/2) -24= $702,000

Answer 2

(#1)

x=1 is the equation for the axis of symmetry....

then -b/2a = 1

-b = 2a

y=-4 is the minimum value, so the vetex is at (1,-4)

and also h=1 and k=-4 for the formula

y = C(x-h)^2+k

the equation is then C(x-1)^2-4

FInally, sincce x=-1 is an intercept,

C(-1-1)^2-4 = 0

C(-2)^2 - 4 = 0

4C - 4 = 0

4C = 4

C=1

The equation is y=(x-1)^2 - 4

= x^2 - 2x + 1 - 4

= x^2 -2x - 3

= (x-3)(x+1)

The x=intercepts are x=3 and x=-1

the vertex is at (3+-1)/2 = 2/2 = 1

The min is (1-3)(1+1) = (-2)(2) = -4

Finally the y-intercept is f(0) = (0-1)^2-4

= (-1)^2-4

1 - 4

= -3

----------------------------------------------------

(2)

Profit = Revenue - Cost

= -2x^2 + 27x - (5x+24)

= -2x^2 + 27x - 5x - 24

= -2x^2 + 22x - 24

Max occurs at

-b/2a = -22/(2(-2)) = -22/-4 = 22/4 = 11/2

and it is -2(11/2)^2 + 22(11/2) - 24 =

-2(121/4) + 22(11/2) - 24 =

-121/2 + 11*11 - 24 =

-121/2 + 121 - 24 =

- 60.5 + 121 - 24=

36.5

So selling 11/2*1000 = 5500 phones makes max profit of 36500