To find the equation of the quadratic function $f(x)$ from the graph provided, we can use the vertex form of a quadratic equation, which is:

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

where $(h, k)$ is the vertex of the parabola.

From the graph:

• The vertex is at $(1, 4)$, so $h = 1$ and $k = 4$.

Now the equation becomes:

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

To find the value of $a$, we can use another point on the parabola. From the graph, another point is $(3, 12)$.

Substitute $(x, y) = (3, 12)$ into the equation:

$12 = a(3 - 1)^2 + 4$

Simplify and solve for $a$:

$12 = a(2)^2 + 4$ $12 = 4a + 4$ $8 = 4a$ $a = 2$

Now, substitute $a = 2$ back into the equation:

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

Thus, the equation of the quadratic function is:

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

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you convert the vertex form of a quadratic equation to the standard form?
2. What is the importance of the vertex in the graph of a quadratic function?
3. How can we determine the axis of symmetry for a quadratic function from its equation?
4. What are the properties of the graph of a quadratic function when the leading coefficient $a$ is negative?
5. How can we find the x-intercepts of a quadratic function?

Tip: Always verify the vertex and another point from the graph to ensure your equation accurately represents the given parabola.