Identify the graph of the following quadratic function.
​f(x)equalsnegative 5​(xplus3​)squared


The given quadratic function is:

\[
f(x) = -5(x + 3)^2
\]

To identify the graph of this function, let's break down its key characteristics:

1. **Vertex form**: The function is in vertex form, \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In this case:
   - \( a = -5 \)
   - \( h = -3 \)
   - \( k = 0 \)

   So, the vertex is \( (-3, 0) \).

2. **Direction of the parabola**: Since \( a = -5 \), which is negative, the parabola opens **downward**.

3. **Stretch/Compression**: The value of \( |a| = 5 \) means the parabola is vertically stretched by a factor of 5 compared to the basic parabola \( y = x^2 \). This makes the graph narrower.

4. **Shift**: The parabola is shifted **3 units to the left** from the origin, as indicated by the \( (x + 3) \) term.

### Summary of the graph:
- **Vertex**: \( (-3, 0) \)
- **Opens downward** (due to \( a = -5 \))
- **Narrower** than the basic parabola
- **No vertical shift** (since \( k = 0 \))

Would you like more details or have any specific questions on this?

Here are 5 related questions to expand your understanding:
1. What is the effect of changing the value of \( a \) on the graph of a parabola?
2. How would the graph change if the function were \( f(x) = 5(x + 3)^2 \)?
3. What would be the equation if the vertex were shifted 2 units up?
4. How does the graph of \( f(x) = -5(x + 3)^2 \) compare to \( f(x) = -x^2 \)?
5. How can you find the x-intercepts of the function \( f(x) = -5(x + 3)^2 \)?

**Tip**: The vertex form of a quadratic function is particularly useful for quickly identifying the vertex and the direction in which the parabola opens.

Identify the graph of the following quadratic function: f(x) = -5(x + 3)^2

Explore how to graph the quadratic function f(x) = -5(x + 3)^2. This problem focuses on identifying the vertex, direction, and stretch of the parabola based on its equation. Suitable for high school students in Grades 9-11.

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