To find the equation of the parabola in vertex form, we follow these steps:

Vertex Form of a Parabola:

The vertex form of a parabola is given by:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the parabola's shape.

Given:

• Vertex $(h, k) = (2, 1)$
• Point $(x, y) = (1, -3)$

Step 1: Plug the vertex into the vertex form equation

Using the vertex $(h, k) = (2, 1)$, the equation becomes:

$f(x) = a(x - 2)^2 + 1$

Step 2: Substitute the point $(x, y) = (1, -3)$ to find $a$

Substitute $x = 1$ and $y = -3$ into the equation:

$-3 = a(1 - 2)^2 + 1$

$-3 = a(-1)^2 + 1$

$-3 = a(1) + 1$

$-3 = a + 1$

Step 3: Solve for $a$

$a = -3 - 1 = -4$

Step 4: Write the final equation

Substitute $a = -4$ back into the equation:

$f(x) = -4(x - 2)^2 + 1$

So, the equation of the parabola in vertex form is:

$f(x) = -4(x - 2)^2 + 1$

Would you like more details or have any other questions?

Related Questions:

1. How do you find the vertex of a parabola given its equation in standard form?
2. What does the value of $a$ represent in the vertex form of a parabola?
3. How can you convert the vertex form of a parabola to standard form?
4. What are the effects of changing $a$ on the shape of the parabola?
5. How do you find the x-intercepts and y-intercept of a parabola in vertex form?

Tip:

When solving problems involving parabolas, always ensure to identify the vertex correctly, as it's crucial for setting up the vertex form equation accurately.