Questions: x y = f(x) -1 6 0 1 1 -4 2 -9 3 -14

Transcript text: \begin{tabular}{|rr|} \hline $\mathbf{x}$ & $\mathbf{y = f}(\mathbf{x})$ \\ \hline-1 & 6 \\ 0 & 1 \\ 1 & -4 \\ 2 & -9 \\ 3 & -14 \\ \hline \end{tabular}

Solution

Solution Steps

To find the relationship between $ x $ and $ y = f(x) $, we can observe the pattern in the table. The values of $ y $ decrease by 5 as $ x $ increases by 1. This suggests a linear relationship of the form $ y = mx + c $. We can use two points from the table to calculate the slope $ m $ and then use one of the points to solve for the intercept $ c $.

Step 1: Identify the Pattern

The table shows a relationship between $ x $ and $ y = f(x) $. Observing the values, we see that as $ x $ increases by 1, $ y $ decreases by 5. This suggests a linear relationship of the form $ y = mx + c $.

Step 2: Calculate the Slope

To find the slope $ m $, we use two points from the table: $(-1, 6)$ and $ (0, 1) $. The formula for the slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values, we get: \[ m = \frac{1 - 6}{0 - (-1)} = \frac{-5}{1} = -5.0 \]

Step 3: Calculate the Intercept

Using the point $(0, 1)$, we can find the intercept $ c $ by substituting into the linear equation $ y = mx + c $: \[ 1 = -5.0 \times 0 + c \implies c = 1.0 \]

Step 4: Formulate the Linear Equation

With $ m = -5.0 $ and $ c = 1.0 $, the linear equation is: \[ y = -5.0x + 1.0 \]