https://file.aiquickdraw.com/mathbot/file/1725402809443jdx7vwif.jpg

The function provided in the image is \( h(x) = (x + 4)^2 \).

This is a quadratic function in vertex form, which can be expanded or analyzed directly.

### Expanding the Function

Let's expand the equation:

\[
h(x) = (x + 4)^2 = (x + 4)(x + 4)
\]

Using the distributive property (FOIL method):

\[
h(x) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16
\]

So, the expanded form of the function is:

\[
h(x) = x^2 + 8x + 16
\]

### Key Features of the Function

- **Vertex**: The vertex of the parabola represented by the function is at \( (-4, 0) \) because the vertex form of a parabola is \( h(x) = (x - h)^2 + k \), where \( (h, k) \) is the vertex. In this case, \( h = -4 \) and \( k = 0 \).
- **Axis of Symmetry**: The axis of symmetry is the vertical line \( x = -4 \).
- **Direction**: The parabola opens upwards because the coefficient of \( x^2 \) is positive.
- **Y-intercept**: The y-intercept occurs when \( x = 0 \). Plugging \( x = 0 \) into the original function:
  
  \[
  h(0) = (0 + 4)^2 = 16
  \]

  So, the y-intercept is \( (0, 16) \).

Would you like more details on any specific part, or do you have any other questions?

### Related Questions:
1. How do you find the vertex of a parabola in standard form?
2. What is the significance of the y-intercept in a quadratic function?
3. How does the value of the coefficient in front of \( x^2 \) affect the shape of the parabola?
4. Can you graph the function \( h(x) = (x + 4)^2 \) and identify its key features?
5. How do you complete the square to rewrite a quadratic function in vertex form?

### Tip:
When dealing with quadratic functions, always check if the function can be factored or if it's easier to analyze in its vertex form, as this can quickly give you key information like the vertex and direction of the parabola.

Explore how to expand and analyze the quadratic function h(x) = (x + 4)^2. Understand its vertex, axis of symmetry, direction, and y-intercept through a step-by-step breakdown suitable for Grades 9-10.

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