To find the equation of the quadratic function $g(x)$ from the given graph, we need to gather some key information from the graph, which includes the vertex and possibly other points.

From the image:

• The vertex is at $(5, -5)$. Since the parabola opens downwards, the equation will have a negative leading coefficient.
• Another point given is $(6, -7)$, which we can use to solve for the unknown coefficient.

Vertex Form of a Quadratic Equation:

The equation of a parabola in vertex form is: $g(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex, and $a$ is the coefficient that determines the width and direction of the parabola.

Step 1: Substituting the Vertex

The vertex is $(5, -5)$, so the equation becomes: $g(x) = a(x - 5)^2 - 5$

Step 2: Use Point $(6, -7)$ to Find $a$

We now substitute the point $(6, -7)$ into the equation to solve for $a$: $-7 = a(6 - 5)^2 - 5$ $-7 = a(1)^2 - 5$ $-7 = a - 5$ $a = -2$

Final Equation

Substitute $a = -2$ back into the equation: $g(x) = -2(x - 5)^2 - 5$

Thus, the equation of the quadratic function is: $g(x) = -2(x - 5)^2 - 5$

Would you like further clarification or more details?

Related Questions:

1. How do you convert from vertex form to standard form for a quadratic equation?
2. What role does the vertex play in determining the shape of a parabola?
3. How do you find the axis of symmetry from the vertex form of a quadratic?
4. Can you always find the equation of a parabola using only the vertex and one other point?
5. What happens if the parabola opens upwards instead of downwards?

Tip:

When identifying the vertex from a graph, always check if the parabola opens upwards or downwards to correctly determine the sign of the leading coefficient.