Question

Determine whether each function is linear or nonlinear. If it is linear, determine the slope. $$ \begin{array}{|rr|} {\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\ \hline-2 & -4 \\ -1 & -3.5 \\ 0 & -3 \\ 1 & -2.5 \\ 2 & -2 \\ \hline \end{array} $$

Short answer

The function is linear with a slope of 0.5.

Step-by-step solution

- Determine if the function is linear

To determine if the function is linear, check if the rate of change between each consecutive pair of points is constant. Calculate the rate of change (or slope) between each pair using the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

- Calculate the slope between the first pair of points

For the points (-2, -4) and (-1, -3.5): \( m_1 = \frac{-3.5 - (-4)}{-1 - (-2)} = \frac{-3.5 + 4}{1} = \frac{0.5}{1} = 0.5 \)

- Calculate the slope between the second pair of points

For the points (-1, -3.5) and (0, -3): \( m_2 = \frac{-3 - (-3.5)}{0 - (-1)} = \frac{-3 + 3.5}{1} = \frac{0.5}{1} = 0.5 \)

- Calculate the slope between the third pair of points

For the points (0, -3) and (1, -2.5): \( m_3 = \frac{-2.5 - (-3)}{1 - 0} = \frac{-2.5 + 3}{1} = \frac{0.5}{1} = 0.5 \)

- Calculate the slope between the fourth pair of points

For the points (1, -2.5) and (2, -2): \( m_4 = \frac{-2 - (-2.5)}{2 - 1} = \frac{-2 + 2.5}{1} = \frac{0.5}{1} = 0.5 \)

- Conclusion

Since the rate of change (or slope) between each pair of consecutive points is constant (0.5), the function is linear. The slope of the function is 0.5.

Key concepts

slope calculation

To understand if a function is linear, one key concept is the slope. The slope measures how steep a line is and is represented by the variable \( m \). The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).
This formula helps us see the rate at which \(y\) changes as \(x\) changes. If we calculate the same value for each pair of points in a table, we can ascertain the nature of the function.

rate of change

When talking about linear functions, the 'rate of change' is simply the slope. This rate remains constant if the function is linear.
For instance, look at the first pair of points (-2, -4) and (-1, -3.5). Calculating the rate of change:
\( m_1 = \frac{-3.5 - (-4)}{-1 - (-2)} = \frac{0.5}{1} = 0.5 \).
Notice that for each consecutive pair of points given in the exercise, the rate of change is always 0.5.
This constancy indicates that the function is linear.

determining linearity

To determine if a function is linear, we check if the rate of change between every pair of consecutive points is the same.
In the exercise, this is done by calculating the slope between each successive pair of points.
If the slope, or rate of change, remains the same, then the function is linear.
In the example provided, all slopes between the pairs of points were 0.5, thus confirming linearity.

difference quotient

The difference quotient is another way to describe the rate of change or slope.
It's calculated using \ \( \frac{f(x+h) - f(x)}{h} \).
This is particularly useful for functions defined by an equation or continuous data.
In our discrete example, we're essentially applying the difference quotient concept to pairs of points.
This method helps us systematically determine the rate of change and check for linearity.