The vertex form of the function g(x)=40x-4x² is g(x) = 4(x + 5)² - 100, where the vertex lies at (-5, -100).
<h3>What is the Equation of a parabola?</h3>
The general equation of a parabola is given as,
y = a(x-h)² + k
where,
(h, k) are the coordinates of the vertex of the parabola in form (x, y);
a defines how narrower is the parabola, and the "-" or "+" that the parabola will open up or down.
As we want to write a function g(x) = 40x + 4x² in vertex form, and we know that the vertex form of a parabola, with vertex at (h,k) is written as,
![f(x) = a(x-h)^2 + k](https://tex.z-dn.net/?f=f%28x%29%20%3D%20a%28x-h%29%5E2%20%2B%20k)
Also, the algebraic identity
is written as,
,
Therefore,
can be written as
![x^2 + 2ax = (x + a)^2 - a^2](https://tex.z-dn.net/?f=x%5E2%20%2B%202ax%20%3D%20%28x%20%2B%20a%29%5E2%20-%20a%5E2)
Further, we try to write the g(x) in the vertex form it can be written as,
![\begin{aligned}g(x) &= 4x^2 + 40x\\\\ &= 4[x^2 + 10x]\\\\ &= 4[(x + 5)^2 - 5^2]\\\\ &= 4(x + 5)^2 - 100\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dg%28x%29%20%26%3D%204x%5E2%20%2B%2040x%5C%5C%5C%5C%20%20%20%20%20%20%20%26%3D%204%5Bx%5E2%20%2B%2010x%5D%5C%5C%5C%5C%20%20%20%20%20%20%20%26%3D%204%5B%28x%20%2B%205%29%5E2%20-%205%5E2%5D%5C%5C%5C%5C%20%20%20%20%20%20%20%26%3D%204%28x%20%2B%205%29%5E2%20-%20100%5C%5C%5Cend%7Baligned%7D)
Hence, the vertex form of the function g(x)=40x-4x² is g(x) = 4(x + 5)² - 100, where the vertex lies at (-5, -100).
Learn more about Parabola:
brainly.com/question/8495268