A downward-facing parabola passes through the points (1,7) ,(3,13), and (9,7) . Find b if the parabola passes through (7,b) .

The equation of a downward-facing parabola is of the form y=a(x−h)2+k, where a is the coefficient of the squared term, h is the horizontal shift of the parabola, and k is the vertical shift of the parabola.

We can find h and k by substituting the known points into the equation. For example, if we substitute the point (1,7), we get:

7=a-2ah+h^2+k

0=-2ah+h^2+6

We can solve this quadratic equation for h to get:

h=3

Substituting the point (3,13), we get:

13=a(3-h)^2+k

13=a-6a+9+k

4=3a+k

k=4-3a

Substituting the point (9,7), we get:

7=a-81a+81+4-3a

7=-78a+85

-78a=78

a=-1

Substituting a=−1, h=3, and k=4−3a into the equation y=a(x−h)2+k, we get:

y=-1(x-3)^2+4-3(-1)

y=-(x^2-6x+9)+7

y=-x^2+6x-2

 

Therefore, if the parabola passes through the point (7,b), then b=−72+6(7)−2=−13​.