Question

A toy rocket is shot up into the air and this is modeled by the function: y=-16x^2+145x+7, Find the vertex and find the zeros of the equation and state whether or not each one is realistic. Show or explain how you found your answer.

Answer 1

Vertex (4.53 , 335.33) Zeroes are -0.048 and 9.11

Explanation:

For find the Vertex also know as the turning point/stationary point and the zeros also known as the x-intercepts of the equation we equate the equation to its complete square form which is

`y=a(x-h)^2+k`

where (h,k) are the vertex point of the equation h corresponds to x coordinate and k corresponds to the y coordinate and a determines whether the graph is a maximum or a minimum. To find whether the graph is a maximum value of a should be < 0 which and if a's value is > 0 then the graph is a minimum. In short

`a<0` maximum (n shaped)

`a > 0` minimum (u shaped

so now we equate, that is

`-16x^2+145x+7=a(x-h)^2+k`

`-16x^2+145x+7=ax^2-2ahx+ah^2+k`

now we compare the coefficients of x^2

-
`-16=a`

now we compare the coefficients of x

`145=-2ah`

`145=-2(-16)h`

`145=32h`

`4.53=h`

now we compare constants

`7=ah^2+k`

`7=(-16)(4.53)^2+k`

`7=-328.33+k`

`335.33=k`

so now we the value of (h , k) that is (4.53 , 335.33)

that is our vertex and since the value of a is less than zero the graph is a maximum.

Now for the zeroes/x-intercepts

we need to find where the graph cuts the x-axis which means if the specific point where the curve cuts the x-axis that point's y coordinate should be zero or more like
`(x,0)`.

and thus this point lies on the curve/equation as well because it satisfies it which means we can put this point into our given equation

`y=-16x^2+145x+7` when we put
`(x,0)` in it y becomes 0 thus changing it to

`-16x^2+145x+7=0`

this becomes a simple quadratic equation where we can use the quadratic formula

`x=-145` ±
`(√((145)^2-4(-16)(7)) )/(-32)`

we get two values of x

`x=(-145+146.54)/(-32)` and
`x=-(-145-146.54)/(-32)`

`x=-0.048` and
`x=9.11` since there 2 values of x means the curve cuts the x-axis at two points we can even confirm our answers by using the desmos graphing calculator as well to check our vertex and zeroes.

Answer 2

I can only tell you the vertex because I don't have a graph to find the zeros. Unless you made a typo in the equation, the vertex is (4.53, 992.18)

Explanation:

To find the vertex when the equation is in standard form, you have to find H and K by using the formula -b/2a. Then once you get the answer, you put that number into where the x's are and solve that, and the total will be the K. After that you just put into vertex form which is f(x)=-16(x-4.53)^2+ 992.18

The H factor will always be opposite of what the sign says in the parenthesis, so the vertex is (4.53, 992.18)