Use the vertex and intercepts to sketch the graph of the quadratic function.  Give the equation of the​ parabola's axis of symmetry. Use the graph to determine the domain and range of the function.
f left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis squared minus 9
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Part 1
Use the graphing tool to graph the function.  Use the vertex and one of the intercepts when drawing the graph.

Let's break this down step by step to help you understand how to sketch the graph of the quadratic function and find the axis of symmetry, domain, and range.

### Given function:
\[
f(x) = (x + 4)^2 - 9
\]
This is a quadratic function in vertex form:
\[
f(x) = a(x - h)^2 + k
\]
Where:
- \( h \) is the x-coordinate of the vertex,
- \( k \) is the y-coordinate of the vertex.

### Step 1: Identify the vertex
The vertex form of the quadratic function \( f(x) = (x + 4)^2 - 9 \) can be rewritten as:
\[
f(x) = (x - (-4))^2 - 9
\]
Thus, the vertex is at \( (-4, -9) \).

### Step 2: Find the axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex. Its equation is given by:
\[
x = h
\]
Since the vertex is \( (-4, -9) \), the axis of symmetry is:
\[
x = -4
\]

### Step 3: Find the intercepts
#### a) **y-intercept**:
The y-intercept occurs when \( x = 0 \). Plug \( x = 0 \) into the equation to find \( y \).
\[
f(0) = (0 + 4)^2 - 9 = 16 - 9 = 7
\]
So the y-intercept is \( (0, 7) \).

#### b) **x-intercepts**:
The x-intercepts occur when \( f(x) = 0 \). Set the equation equal to zero and solve for \( x \):
\[
(x + 4)^2 - 9 = 0
\]
\[
(x + 4)^2 = 9
\]
Taking the square root of both sides:
\[
x + 4 = \pm 3
\]
So:
\[
x + 4 = 3 \quad \text{or} \quad x + 4 = -3
\]
This gives:
\[
x = -1 \quad \text{or} \quad x = -7
\]
Thus, the x-intercepts are \( (-1, 0) \) and \( (-7, 0) \).

### Step 4: Graph the parabola
- The vertex is \( (-4, -9) \).
- The parabola opens upwards since the coefficient of \( (x + 4)^2 \) is positive.
- The y-intercept is \( (0, 7) \).
- The x-intercepts are \( (-1, 0) \) and \( (-7, 0) \).
- The axis of symmetry is \( x = -4 \).

### Step 5: Determine the domain and range
#### a) **Domain**:
Since the function is a quadratic, its domain is all real numbers:
\[
\text{Domain}: (-\infty, \infty)
\]

#### b) **Range**:
The vertex \( (-4, -9) \) represents the minimum point because the parabola opens upwards. Therefore, the range is:
\[
\text{Range}: [-9, \infty)
\]

Would you like to see the graph of this function, or need further details?

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Here are 5 related questions:
1. How do you convert a quadratic equation from standard form to vertex form?
2. What are the properties of a parabola with a positive leading coefficient?
3. How can you calculate the distance between the x-intercepts?
4. How does changing the value of \( h \) and \( k \) in the vertex form affect the graph?
5. What is the significance of the axis of symmetry in quadratic functions?

**Tip:** The vertex form of a quadratic equation is especially useful when you want to quickly identify the vertex and graph the function efficiently.

Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the domain and range of the function. f(x) = (x + 4)^2 - 9

Learn how to graph the quadratic function f(x) = (x + 4)^2 - 9 by using its vertex, axis of symmetry, and intercepts. This tutorial covers key concepts including finding the domain and range, and is suitable for students in Grades 9-11.

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