The given function is:

$g(a) = \frac{1}{(a + 3)^2} + \frac{1}{a + 3} - 2$

To study this function using a quadratic model, we first need to rewrite and analyze the behavior of $g(a)$ near certain points, identify its features, and then approximate it with a quadratic expression.

Step 1: Rewrite the function for analysis

First, let $u = a + 3$, so that $g(a)$ becomes:

$g(u) = \frac{1}{u^2} + \frac{1}{u} - 2$

Step 2: Analyze the function for approximation

Let's expand $g(u)$:

$g(u) = \frac{1}{u^2} + \frac{1}{u} - 2$

Now, observe that as $u$ increases, $\frac{1}{u^2}$ and $\frac{1}{u}$ become smaller. To model this behavior quadratically, we look at values of $u$ close to a reference point (for example, $u = 1$, which corresponds to $a = -2$).

Step 3: Approximate using a quadratic model

To find a quadratic approximation, we can expand the function $g(u)$ around a point using a Taylor series expansion or use known points to fit a quadratic curve of the form:

$g(u) \approx p(u) = A(u - u_0)^2 + B(u - u_0) + C$

Where $u_0$ is the reference point and $A$, $B$, and $C$ are constants to be determined based on the properties of the original function.

Step 4: Finding maximum, minimum, and zero points

• Maximum/Minimum: The vertex of the quadratic model $p(u)$ will give us the maximum or minimum values of the function.
• Zeros: If the quadratic model crosses the x-axis, that will correspond to the zeros of the function.

Conclusion:

To determine the exact quadratic approximation, we would either expand $g(u)$ at a specific point or fit the curve based on derivative analysis. However, based on the shape of the function $g(a)$, it seems that this function has a minimum but does not have zeros because both terms $\frac{1}{u^2}$ and $\frac{1}{u}$ are always positive.

Would you like a detailed quadratic expansion or help with plotting the function to determine these values more precisely?

Here are 5 related questions to consider:

1. How can we expand functions using a Taylor series?
2. What are the steps to convert a function into a quadratic model using approximation?
3. How do we find the vertex of a quadratic model in standard form?
4. What is the importance of choosing the reference point $u_0$ when approximating?
5. How do we compute zeros of a quadratic function?

Tip: When modeling non-linear functions, quadratic approximations are useful for understanding local behavior near specific points.