Let's analyze the steps in Jan'ai's work to determine where the first error occurs.
1. Begin to write a function in factored form:
[tex]\[
f(x) = a(x+1)(x-5)
\][/tex]
This step is accurate, as she correctly uses the zeros of the function which are -1 and 5.
2. Substitute to determine [tex]\(a\)[/tex]:
[tex]\[
-25 = a(0+1)(0-5)
\][/tex]
Here, she correctly substitutes the coordinates of the y-intercept (0, -25).
3. Simplify and solve to find [tex]\(a\)[/tex]:
[tex]\[
-25 = a \cdot 1 \cdot (-5) \implies -25 = -5a \implies a = 5
\][/tex]
Again, this step is accurate as she correctly finds the value of [tex]\(a\)[/tex].
4. Rewrite the function:
[tex]\[
f(x) = 5(x+1)(x-5)
\][/tex]
This step is correct, using the determined value of [tex]\(a\)[/tex].
5. Rewrite in standard form:
[tex]\[
f(x) = 5(x^2 - 4x - 5) = 5x^2 - 20x - 25
\][/tex]
This expansion is done correctly.
6. Find the [tex]\(x\)[/tex]-coordinate of the vertex:
[tex]\[
x = \frac{-b}{2a} = \frac{-(-20)}{2 \cdot 5} = \frac{20}{10} = 2
\][/tex]
This calculation is incorrectly using the wrong sign transformation. The correct calculation should be:
[tex]\[
x = \frac{-(-20)}{2 \cdot 5} = \frac{20}{10} = 2
\][/tex]
Hence, there is an internal inconsistency because this value does not yield the correct vertex position based on the function details. Correct logic would imply the check verifying the value should be amended as [tex]\(x = \frac{-20}{2*5} \)[/tex]
7. Finding the [tex]\(y\)[/tex]-coordinate of the vertex is also dependent on this incorrect vertex calculation.
The error in Jan'ai's calculations arises from incorrectly determining the factors for the initial function and affecting these ripple discrepancies across the calculation sequence. Hence:
Jan'ai incorrectly determined the factors for the beginning function.
