To determine the equation that describes the function presented in the table, we can first observe how \(y\) changes as \(x\) increases. Let's analyze the values:
\[ \begin{align*} \text{When } x = 0, & \quad y = -5 \ \text{When } x = 1, & \quad y = -9 \ \text{When } x = 2, & \quad y = -13 \ \text{When } x = 3, & \quad y = -17 \ \end{align*} \]
Now, we notice that as \(x\) increases by 1, \(y\) decreases by 4. This suggests a linear relationship.
Let's express \(y\) in terms of \(x\). The general form of a linear equation is:
\[ y = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept. Since \(y\) decreases by 4 for each increase of 1 in \(x\), the slope \(m\) is \(-4\).
To find the y-intercept \(b\), we can use the value of \(y\) when \(x = 0\):
\[ y = -5 = -4(0) + b \implies b = -5 \]
Now, we can write the equation for the function:
\[ y = -4x - 5 \]
In function notation, we can express this as:
\[ f(x) = -4x - 5 \]
Therefore, the equation that correctly describes the function in the table is:
\[ f(x) = -4x - 5 \]