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.