The first error in Jan'ai's work in determining the considered function is given by: Option D: She incorrectly determined the x-coordinate of the vertex.
<h3>What are the coordinates of vertex for a quadratic function?</h3>
For a quadratic function of the form
, its vertex form is obtained as:
![y = ax^2 + bx + c\\y =a(x^2 + bx/a) + c\\y = a(x^2 + 2(b/2a)x + (b/2a)^2 -(b/2a)^2 )+ c\\y = a(x^2 + 2(b/2a)x + (b/2a)^2) -a \times (b/2a)^2 + c\\\\y = a(x+b/2a)^2 - a \times (b/2a)^2 + c](https://tex.z-dn.net/?f=y%20%3D%20ax%5E2%20%2B%20bx%20%2B%20c%5C%5Cy%20%20%3Da%28x%5E2%20%2B%20bx%2Fa%29%20%2B%20c%5C%5Cy%20%3D%20a%28x%5E2%20%2B%202%28b%2F2a%29x%20%2B%20%28b%2F2a%29%5E2%20-%28b%2F2a%29%5E2%20%20%29%2B%20c%5C%5Cy%20%3D%20a%28x%5E2%20%2B%202%28b%2F2a%29x%20%2B%20%28b%2F2a%29%5E2%29%20-a%20%5Ctimes%20%28b%2F2a%29%5E2%20%2B%20c%5C%5C%5C%5Cy%20%3D%20a%28x%2Bb%2F2a%29%5E2%20-%20a%20%5Ctimes%20%28b%2F2a%29%5E2%20%2B%20c)
For the form
, the vertex has coordinates (h, k)
Thus, for the obtained equation
, we get the coordinates of vertex as:
, ![k = c - a\times(b/2a)^2](https://tex.z-dn.net/?f=k%20%3D%20c%20-%20a%5Ctimes%28b%2F2a%29%5E2)
Thus, the coordinates of vertex of
is:
![(h,k) = (-b/2a, c - a \times (b/2a)^2 )](https://tex.z-dn.net/?f=%28h%2Ck%29%20%3D%20%28-b%2F2a%2C%20c%20-%20a%20%5Ctimes%20%28b%2F2a%29%5E2%20%29)
The missing steps of work of Jan'ai are:
- Begin to write a function in factored form.
![f(x) = a(x+1)(x-5)](https://tex.z-dn.net/?f=f%28x%29%20%3D%20a%28x%2B1%29%28x-5%29)
- Substitute x = 0, y = f(x) = -25 to determine a.
![-25 = a(0+1)(0-5)](https://tex.z-dn.net/?f=-25%20%3D%20a%280%2B1%29%280-5%29)
- Simplify and solve to find a.
![a=5](https://tex.z-dn.net/?f=a%3D5)
- Rewrite the function.
![f(x) = 5(x+1)(x-5)](https://tex.z-dn.net/?f=f%28x%29%20%3D%205%28x%2B1%29%28x-5%29)
- Rewrite in standard form.
![f(x) = 5x^2-20x-25](https://tex.z-dn.net/?f=f%28x%29%20%3D%205x%5E2-20x-25)
- Find the x-coordinate of the vertex.
![x = -20/2(5) = -20/10; x = -2](https://tex.z-dn.net/?f=x%20%3D%20-20%2F2%285%29%20%3D%20-20%2F10%3B%20x%20%3D%20-2)
- Find the y-coordinate of the vertex.
![y = 5x^2-20x-25](https://tex.z-dn.net/?f=y%20%3D%205x%5E2-20x-25)
![y = 5(-2)^2-20(-2)-25](https://tex.z-dn.net/?f=y%20%3D%205%28-2%29%5E2-20%28-2%29-25)
![y = 35](https://tex.z-dn.net/?f=y%20%3D%2035)
so (-2,35) is the coordinate of the vertex, which denotes the minimum.
So, as we see, in the 5th step, Jan'ai had the quadratic function
,
Comparing this to
, we get a = 5, b = -20, c = -25
The vertex's x-coordinate will be on -b/2a = -(-20)/ 2(5) = 20/10 = 2
But Jan'ai didn't putted that negative sign before b. in the 6th step.
Thus, the first error in Jan'ai's work in determining the considered function is given by: Option D: She incorrectly determined the x-coordinate of the vertex.
Learn more about the vertex form of a quadratic equation here:
brainly.com/question/9912128