From the graph you have uploaded, it appears that the quadratic function is a parabola that opens upwards. Let's break this down step by step:

Step 1: Identify Key Points on the Graph

• The vertex of the parabola is at $(0, -4)$, which means the lowest point of the parabola is at $y = -4$.
• The x-intercepts (where the graph crosses the x-axis) are $(-2, 0)$ and $(2, 0)$.

Step 2: Write the Factored Form of the Equation

A quadratic function with roots (x-intercepts) at $x = -2$ and $x = 2$ can be written in factored form as: $y = a(x + 2)(x - 2)$ Since the vertex is at $(0, -4)$, we can substitute this point to solve for $a$.

Step 3: Solve for $a$

Substitute $(0, -4)$ into the factored equation: $-4 = a(0 + 2)(0 - 2)$ $-4 = a(2)(-2)$ $-4 = -4a$ Solving for $a$: $a = 1$

Thus, the factored form of the equation is: $y = (x + 2)(x - 2)$ or simply: $y = (x^2 - 4)$

Step 4: Write the Standard Form of the Equation

To convert the factored form into standard form, expand the expression: $y = (x + 2)(x - 2)$ Using the difference of squares formula: $y = x^2 - 4$

Final Answer:

• Factored form: $y = (x + 2)(x - 2)$
• Standard form: $y = x^2 - 4$

Would you like further details on this problem or another aspect explained?

Here are some related questions you might consider:

1. How do you find the vertex of a parabola using its equation?
2. What is the significance of the axis of symmetry in quadratic functions?
3. How do you convert a quadratic function from standard form to vertex form?
4. What role does the value of $a$ play in the shape of the parabola?
5. How can you determine if a parabola opens upwards or downwards?

Tip: Factoring is often the simplest method for finding x-intercepts of a quadratic function, but completing the square or using the quadratic formula are also reliable alternatives!