To analyze the function $g(x) = 0.35x^2 - 0.7x - 9$, let's begin by rewriting it in vertex form, which is $g(x) = a(x - h)^2 + k$. Here:

1. $a = 0.35$
2. We need to complete the square to find $h$ and $k$.

Part 1: Convert to Vertex Form

Starting with: $g(x) = 0.35x^2 - 0.7x - 9$

Step 1: Factor out $0.35$ from the $x$ terms:

$g(x) = 0.35(x^2 - 2x) - 9$

Step 2: Complete the square inside the parentheses.

For $x^2 - 2x$, we add and subtract $1$ (because $\left(\frac{-2}{2}\right)^2 = 1$):

$g(x) = 0.35(x^2 - 2x + 1 - 1) - 9$

$= 0.35((x - 1)^2 - 1) - 9$

Step 3: Distribute $0.35$ and simplify:

$= 0.35(x - 1)^2 - 0.35 - 9$

$= 0.35(x - 1)^2 - 9.35$

So, the vertex form is: $g(x) = 0.35(x - 1)^2 - 9.35$

The correct choice is Option C.

Part 2: Intercepts

(A) X-Intercepts

To find the $x$-intercepts, set $g(x) = 0$: $0 = 0.35(x - 1)^2 - 9.35$

Solving for $x$: $0.35(x - 1)^2 = 9.35$

$(x - 1)^2 = \frac{9.35}{0.35} = 26.71$

$x - 1 = \pm \sqrt{26.71} \approx \pm 5.17$

$x \approx 1 \pm 5.17$

So, the $x$-intercepts are approximately: $x \approx 6.17 \quad \text{and} \quad x \approx -4.17$

(B) Y-Intercept

For the $y$-intercept, set $x = 0$: $g(0) = 0.35(0)^2 - 0.7(0) - 9 = -9$

So, the $y$-intercept is $(0, -9)$.

Part 3: Vertex

From the vertex form $g(x) = 0.35(x - 1)^2 - 9.35$, the vertex is: $(1, -9.35)$

Part 4: Maximum or Minimum

Since the coefficient of $x^2$ (i.e., $0.35$) is positive, the parabola opens upwards, meaning it has a minimum at the vertex.

The minimum value of $g(x)$ is $-9.35$, which occurs at $x = 1$.

Part 5: Range

Since the minimum value of $g(x)$ is $-9.35$ and the function opens upwards, the range is: $[-9.35, \infty)$

Summary of Answers

1. Vertex Form: $g(x) = 0.35(x - 1)^2 - 9.35$ (Option C)
2. X-Intercepts: Approximately $x \approx 6.17$ and $x \approx -4.17$
3. Y-Intercept: $(0, -9)$
4. Vertex: $(1, -9.35)$
5. Minimum Value: $-9.35$ at $x = 1$
6. Range: $[-9.35, \infty)$

Would you like any additional explanations on these steps?

Related Questions

1. How do you complete the square for a quadratic function?
2. Why does the sign of the leading coefficient affect the direction of the parabola?
3. How can you find intercepts directly from the standard form of a quadratic function?
4. How is the vertex used to determine the maximum or minimum of a parabola?
5. What are the characteristics of a quadratic function's graph that help define its range?

Tip

For any quadratic function $f(x) = ax^2 + bx + c$, completing the square is a reliable method to rewrite it in vertex form and easily identify the vertex and other key properties.