graph the quadratic function given in vertex form. $$ f\left(x\right)=2\left(x+5\right)^2-1 $$

To graph the quadratic function $ f(x) = 2(x+5)^2 - 1 $, follow these steps:

### Step 1: Identify the vertex
The quadratic function is in **vertex form**, which is given by:
\[
f(x) = a(x-h)^2 + k
\]
Here, the vertex is at $ (h, k) $. 

For $ f(x) = 2(x+5)^2 - 1 $, you can identify:
- $ h = -5 $
- $ k = -1 $

Thus, the vertex is $ (-5, -1) $.

### Step 2: Determine the direction of the parabola
The coefficient $ a = 2 $ is positive, so the parabola opens **upwards**. Since $ a = 2 $, the parabola is vertically stretched, meaning it is narrower than the standard parabola $ f(x) = (x-h)^2 + k $.

### Step 3: Find additional points
You can calculate a few more points by choosing $ x $-values around the vertex:

- For $ x = -4 $: 
  \[
  f(-4) = 2(-4+5)^2 - 1 = 2(1)^2 - 1 = 1
  \]
  So, the point is $ (-4, 1) $.

- For $ x = -6 $:
  \[
  f(-6) = 2(-6+5)^2 - 1 = 2(-1)^2 - 1 = 1
  \]
  So, the point is $ (-6, 1) $.

- For $ x = 0 $:
  \[
  f(0) = 2(0+5)^2 - 1 = 2(5)^2 - 1 = 49
  \]
  So, the point is $ (0, 49) $.

### Step 4: Plot the graph
1. Plot the vertex $ (-5, -1) $.
2. Plot the additional points such as $ (-4, 1) $, $ (-6, 1) $, and $ (0, 49) $.
3. Draw the parabola, making sure it is symmetric around $ x = -5 $ and opening upwards.

Would you like me to generate a graph for this function or explain any step further?

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Here are 5 related questions:
1. How does changing the coefficient of $ a $ affect the shape of the parabola?
2. What is the axis of symmetry for the given function?
3. Can you find the y-intercept for the function?
4. What happens to the graph if we change $ k $ in the vertex form?
5. How would the graph look if $ a $ was negative instead of positive?

**Tip:** The vertex form of a quadratic function is helpful because it directly provides the vertex, making it easier to graph.

Graph the quadratic function given in vertex form. $$ f\left(x\right)=2\left(x+5\right)^2-1 $$

Learn how to graph the quadratic function f(x) = 2(x + 5)^2 - 1 step by step. Identify the vertex, calculate additional points, and understand the symmetry of the parabola. Suitable for students in Grades 9-12, this lesson covers graphing parabolas in vertex form and analyzing their properties.

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