The graph shows a quadratic function, and we are tasked with finding its equation in standard form.

Step 1: General form of a quadratic function

The standard form of a quadratic function is: $y = ax^2 + bx + c$ We can also use the vertex form of a quadratic function to find the equation, which is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Step 2: Identify the vertex

From the graph, the vertex is clearly $(2, -6)$. So, $h = 2$ and $k = -6$.

Thus, the equation becomes: $y = a(x - 2)^2 - 6$

Step 3: Use another point to find "a"

Now, we need to find the value of $a$. We can use another point on the graph to do this. The point $(5, 3)$ lies on the parabola. Plug this point $(x, y) = (5, 3)$ into the vertex form of the equation:

$3 = a(5 - 2)^2 - 6$ $3 = a(3)^2 - 6$ $3 = 9a - 6$ $9 = 9a$ $a = 1$

Step 4: Write the equation in vertex form

Now that we know $a = 1$, the equation in vertex form is: $y = (x - 2)^2 - 6$

Step 5: Convert to standard form

Finally, let's expand this equation into standard form: $y = (x - 2)^2 - 6$ $y = (x^2 - 4x + 4) - 6$ $y = x^2 - 4x - 2$

Thus, the equation of the quadratic function in standard form is: $y = x^2 - 4x - 2$

Would you like more details on any part of this process or have any questions?

Related Questions:

1. How do you find the vertex of a parabola from its equation in standard form?
2. What are the differences between vertex form and standard form of a quadratic equation?
3. How can you determine the direction of a parabola from its equation?
4. How do you find the x-intercepts of a quadratic equation?
5. How does changing the value of "a" affect the shape of a parabola?

Tip:

When working with quadratic functions, always check the vertex and another point on the graph to accurately determine the function's parameters.