Graph the function f(x)=
–
1
2
(x+3)2.
Plot the vertex. Then plot another point on the parabola.

The function given is:

\[
f(x) = -\frac{1}{2}(x + 3)^2
\]

This is a quadratic function in vertex form:

\[
f(x) = a(x - h)^2 + k
\]

where \(a = -\frac{1}{2}\), \(h = -3\), and \(k = 0\). The vertex of this parabola is at \((h, k) = (-3, 0)\).

### Key Features:
1. The negative sign in front of the \(a\) value indicates the parabola opens downward.
2. The vertex is at \((-3, 0)\).
3. The factor of \(-\frac{1}{2}\) affects the "width" of the parabola, making it wider than a standard parabola.

### Steps to Plot the Graph:

1. **Vertex**: Plot the vertex at \((-3, 0)\).
2. **Another Point**: To find another point, let's choose \(x = -2\) and substitute into the equation:
   
   \[
   f(-2) = -\frac{1}{2}((-2 + 3)^2) = -\frac{1}{2}(1^2) = -\frac{1}{2}
   \]
   
   So, another point is \((-2, -\frac{1}{2})\).

### Graph of the function:

- Plot the vertex at \((-3, 0)\).
- Plot the point \((-2, -\frac{1}{2})\).
- Reflect the point \((-2, -\frac{1}{2})\) across the axis of symmetry (which is \(x = -3\)) to find another point, which would be \((-4, -\frac{1}{2})\).
  
### Symmetry:
Since this is a parabola, it is symmetric about the vertical line \(x = -3\).

Would you like me to graph this function for you? If so, I can create and upload the plot.

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### Follow-up questions:
1. How does the value of \(a\) in the equation affect the width of the parabola?
2. What happens to the graph if the vertex is moved to a different point?
3. How does the negative sign in front of \(a\) impact the direction of the parabola?
4. What is the axis of symmetry for this quadratic function?
5. How do you find the y-intercept of the parabola?

**Tip**: When graphing quadratic functions, always start by identifying the vertex and the direction in which the parabola opens (up or down).

Graph the function f(x)= -1/2 (x+3)^2. Plot the vertex. Then plot another point on the parabola.

Learn how to graph the quadratic function f(x) = -1/2 (x+3)^2 by identifying its vertex and plotting another point on the parabola. This problem covers key concepts like the vertex form of a quadratic equation, the axis of symmetry, and understanding the width of the parabola. Ideal for students in Grades 9-11.

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