Determine whether the function is linear or nonlinear. If...

Determine whether the function is linear or nonlinear. If linear, determine the equation that defines $y=f(x)$.
$$
\begin{array}{|rr|}
\hline {\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-6 & -3 \\
-3 & -4 \\
0 & -5 \\
3 & -6 \\
6 & -7 \\
\hline
\end{array}
$$

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Key Concepts

Linear Function

A linear function is defined by an equation of the form y = mx + b, where the graph of the function is a straight line. This means that for every unit increase in x, the value of y increases or decreases by a constant amount, which is the slope of the line.

Slope

The slope (m) represents the constant rate of change of the function and is calculated by dividing the change in y by the change in x between any two points on the line. A constant slope is a defining characteristic of a linear function.

Y-intercept

The y-intercept (b) is the value of y when x is zero. It indicates where the line crosses the y-axis and is a crucial component in the formulation of the linear equation.

Testing for Linearity

To determine if a function is linear, one can check if the rate of change (slope) between any two pairs of points is consistent. This can be done by comparing the differences in the y-values relative to the corresponding differences in the x-values, or by plotting the points to see if they form a straight line.

Equation of a Line

Once linearity is confirmed, the next step is to determine the equation of the line. This is typically done by using the slope-intercept form, y = mx + b, where m and b are identified from the data or graph. This equation provides a complete description of the linear relationship between x and y.

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