To determine the function that represents the graph of the parabola given the [tex]\(y\)[/tex]-intercept of 1 and a vertex at [tex]\((1, 0)\)[/tex], let's break down the problem logically and methodically.
First, we know that the vertex form of a parabola is generally represented as:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. For our problem, the vertex [tex]\((h, k)\)[/tex] is given as [tex]\((1, 0)\)[/tex]. Substituting [tex]\(h\)[/tex] and [tex]\(k\)[/tex] into the vertex form, we have:
[tex]\[ f(x) = a(x - 1)^2 \][/tex]
Next, we'll use the information about the [tex]\(y\)[/tex]-intercept. The [tex]\(y\)[/tex]-intercept means the value of the function when [tex]\(x = 0\)[/tex].
When [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = a(0 - 1)^2 = a(1)^2 = a \][/tex]
We are given that the [tex]\(y\)[/tex]-intercept is 1. Thus:
[tex]\[ f(0) = 1 \implies a = 1 \][/tex]
Therefore, the equation of the parabola is:
[tex]\[ f(x) = (x - 1)^2 \][/tex]
Now we will check this equation against the given answer choices:
A. [tex]\( f(x) = (x - 1)^2 \)[/tex]
B. [tex]\( f(x) = (x + 1)^2 \)[/tex]
C. [tex]\( f(x) = -1(x - 1)^2 \)[/tex]
D. [tex]\( f(x) = -1(x + 1)^2 \)[/tex]
Since option A matches our derived function, it is the correct answer.
Thus, the correct function that represents the graph is:
[tex]\[ \boxed{f(x) = (x - 1)^2} \][/tex]
Option A is the correct answer.
