https://file.aiquickdraw.com/mathbot/file/1725820616110gch2mtw4.jpg

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!

Write the equations of this quadratic function in factored and standard forms.

This tutorial explains how to derive the factored and standard forms of a quadratic function based on its graph. The function has x-intercepts at -2 and 2, and a vertex at (0, -4).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation