solve this---13
Chart Description:
The image contains a graph in the xy-plane.
- Type: Parabola.
- Axes: X-axis and Y-axis are labeled. The origin O is marked.
- Curve: A parabola opening upwards is shown, labeled as $y = f(x)$.
- Points: Three points on the parabola are marked and labeled with their coordinates: (2, 5), (3, 1), and (4, 5). The point (3, 1) appears to be the vertex of the parabola.
Question Stem:
The graph of the function $f$ in the $xy$-plane above is a parabola. Which of the following defines $f$?
Options:
A) $f(x) = 4(x-3)^2 + 1$
B) $f(x) = 4(x+3)^2 + 1$
C) $f(x) = (x-3)^2 + 1$
D) $f(x) = 3(x+3)^2 + 1$
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In this problem, we need to find the equation of a parabola that passes through three given points: (2, 5), (3, 1), and (4, 5). Looking at the graph, we can see that the point (3, 1) appears to be the vertex of the parabola. We'll use the vertex form of a parabola equation to solve this problem.
To find the equation of this parabola, we'll use the vertex form. First, we identify the vertex of the parabola, which is the lowest point at (3, 1). The vertex form of a parabola is f(x) = a(x-h)^2 + k, where (h,k) is the vertex. Substituting our vertex coordinates, we get f(x) = a(x-3)^2 + 1. Now we need to find the value of a.
Now we need to find the value of parameter 'a'. We'll use one of the given points, (2, 5), and substitute it into our equation. We get 5 equals a times (2 minus 3) squared plus 1. Simplifying, we get 5 equals a times (-1) squared plus 1, which is 5 equals a times 1 plus 1, or 5 equals a plus 1. Solving for a, we get a equals 4. Therefore, the final equation of the parabola is f(x) equals 4 times (x minus 3) squared plus 1.