Two questions on parabola equations and turning points

Question

1. Find B if the parabola y = B(3 – x)(x + 5) has a turning point at (–1,4)

 

 

2. Find equation

Turning point at (1,5) and y-intercept at (0,3)

Answer 1

1)

 

The turning point is the vertex of the parabola.  The vertex is also a point on the parabola.  So we plug in x=-1 and y=4 into the equation.  That will allow us to solve for B.

 

4 = B(3 - (-1))(-1 + 5)

4 = B(4)(4)

4 = 16B

 

1/4 = B

 

 

2)

 

For this one, we can use the vertex form of the parabola

 

y = a(x - h)² + k

 

where:

a is the coefficient of the x² term

the vertex has coordinate (h, k)

 

Since we already know the vertex and y-intercept, we can plug in these values into the equation:

 

x = 0

y = 3

h = 1

k = 5

 

and solve for a.

 

3 = a(0 - 1)² + 5

3 = a + 5

 

-2 = a

 

 

The equation is then

 

y = -2(x - 1)² + 5

 

Expanding, we get

 

y = -2(x² - 2x + 1) + 5

 

y = -2x² + 4x + 3