suppose f is a quadratic function with the vertex (-3,-5) and x-intercepts x=-2 and x=-4. Find its y-intercept.

Question

Make sure the answer is supported by reason

Answer 1

Quadratic function so the graph would be a parabola

 

Vertex = ( -3, -5 )  x - intercepts  -2, -4

 

Using the Graphing Form for equation of parabola:

 

y = a ( x - h )²  + k ;  vertex is (h,K) ;  a = constriction of curve related to y = x²

 

Therefor:                y = a ( x - (-3))² - 5

 

Using pt. (-2 , 0):   0 = a ( -2 + 3)² -5

 

                              5 = a

 

Equation of parabola is:    y = 5 ( x - (-3))² - 5

 

Letting x = 0                   y = 5 (9) - 5

 

                                       y = 40 

 

Y-intercept is (0,40)

 

Answer 2

Since the curve intersects the x-axis (y = 0) at x = -2 and x = -4, these values are roots of the quadratic equation, so it has the form

 

f(x) = A(x + 2)(x + 4)

 

where A is a constant.  Use the given point (-3,-5) to find A.

 

-5 = A(-3 + 2)(-3 + 4) = A(-1)(1) = -A

 

A = 5

 

Then

 

f(x) = 5(x + 2)(x + 4)

 

Finally, the y-intercept is

 

f(0) = 5*2*4 = 40

 

with coordinates (0,40).