Quadratic Functions problem

Question

I have an equation of "a1 released his ballon from the 10-yard line, and it landed at the 16-yard line, if the ball reached a height of 27 years, what equation represents the path of his toss?I need help finding the equation

Answer 1

Draw a sketch:

Doesn't it make sense that the high spot must occur in the middle, between 10 and 16 so at the 13 yard line?

So you have three points, (10, 0), (13, 27), and (16, 0)

You can use the factored form of a quadratic:

y = a(x - 10)(x - 16) because you have the two x-intercepts. To find the value of "a", you can use the other point (13, 27):

27 = a(13 - 10)(13 - 16)

27 = a(3)(-3)

27 = -9a

a = -3

So the factored form of the quadratic equation is y = -3(x - 10)(x - 16)

If you multiply this out, you get -3(x² - 26x + 160) = -3x² + 78x - 480

Another way to do the problem is to use the vertex form y = a(x - h)² + k and use the vertex (h, k) where h = 13 and k = 27:

y = a(x - 13)² + 27 then to calculate "a", plug in either (10, 0) or (16, 0). I'll use (10, 0):

0 = a(10 - 13)² + 27

0 = a(-3)² + 27

0 = 9a + 27

-27 = 9a

a = -3

So the equation is y = -3(x - 13)² + 27.

If you multiply this equation out, you also get y = -3x² + 78x - 480