Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.