To determine the function representing the relationship, let's first calculate the slope $m$ of the linear function using two points from the table. The slope $m$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Choose two points, say $(-7.5, 12)$ and $(-3.5, 0)$:

$m = \frac{0 - 12}{-3.5 - (-7.5)} = \frac{-12}{-3.5 + 7.5} = \frac{-12}{4} = -3$

Now that we have the slope $m = -3$, the equation of the line is in the form:

$y = mx + b$

Substitute one of the points, say $(-7.5, 12)$, into the equation to solve for $b$:

$12 = -3(-7.5) + b$ $12 = 22.5 + b$ $b = 12 - 22.5 = -10.5$

Thus, the equation of the line is:

$y = -3x - 10.5$

The corresponding function is:

$h(x) = -3x - 10.5$

Final Answer:

C. $h(x) = -3x - 10.5$

Would you like a detailed explanation or graph for better visualization?

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Related Questions:

1. How do you calculate the slope of a linear function from a table?
2. What is the importance of the $y$-intercept ($b$) in a linear equation?
3. How can you verify if a point lies on a given linear equation?
4. Why does the slope remain constant in a linear function?
5. Can the equation of a linear function be written in different forms?

Tip:

To quickly verify a linear equation, substitute at least two points from the table into the equation. If both satisfy the equation, it is correct!