To find the equation of a parabola, given its vertex and a point through which it passes, we can use the vertex form of the parabola equation. The vertex form of a parabola is written as:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(a\)[/tex] is a coefficient that determines the width and the direction of the parabola.
For the given problem:
- The vertex of the parabola is [tex]\((1, 0)\)[/tex]. Therefore, [tex]\(h = 1\)[/tex] and [tex]\(k = 0\)[/tex].
- The parabola passes through the point [tex]\((3, 4)\)[/tex]. This means when [tex]\(x = 3\)[/tex], [tex]\(y = 4\)[/tex].
Substitute the vertex [tex]\((h, k)\)[/tex] into the vertex form of the equation:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Since [tex]\(h = 1\)[/tex] and [tex]\(k = 0\)[/tex], the equation becomes:
[tex]\[ y = a(x - 1)^2 + 0 \][/tex]
or simply:
[tex]\[ y = a(x - 1)^2 \][/tex]
Next, substitute the point [tex]\((3, 4)\)[/tex] into this equation to find the value of [tex]\(a\)[/tex]:
[tex]\[ 4 = a(3 - 1)^2 \][/tex]
Simplify inside the parentheses:
[tex]\[ 4 = a(2)^2 \][/tex]
This simplifies to:
[tex]\[ 4 = 4a \][/tex]
Now, solve for [tex]\(a\)[/tex]:
[tex]\[ a = 1 \][/tex]
With the value of [tex]\(a\)[/tex] determined, substitute it back into the equation:
[tex]\[ y = 1(x - 1)^2 \][/tex]
So, the equation of the parabola is:
[tex]\[ y = (x - 1)^2 \][/tex]
Therefore, the equation of the parabola with vertex [tex]\((1, 0)\)[/tex] that passes through the point [tex]\((3, 4)\)[/tex] is:
[tex]\[ y = (x - 1)^2 \][/tex]
