To determine the correct equation that describes the function represented by the table, we can first analyze the given points:
\[ \begin{align*} (0, -5) \ (1, -9) \ (2, -13) \ (3, -17) \ \end{align*} \]
We can observe that as \( x \) increases, \( y \) decreases.
To find a linear equation of the form \( y = mx + b \), we need to determine the slope \( m \) and the y-intercept \( b \).
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Calculate the slope \( m \): Using two points, for example, \( (0, -5) \) and \( (1, -9) \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-9 - (-5)}{1 - 0} = \frac{-9 + 5}{1} = \frac{-4}{1} = -4 \]
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Find the y-intercept \( b \): We can use the y-value when \( x = 0 \) (which is -5) in the equation \( y = mx + b \): \[ -5 = -4(0) + b \ b = -5 \]
Now we have both \( m \) and \( b \). The equation of the function is:
\[ y = -4x - 5 \]
Using function notation, we write:
\[ f(x) = -4x - 5 \]
Thus, the correct equation that describes the function in the table is:
\[ f(x) = -4x - 5 \]