$x$ | $f ( x )$ |
-4 | -10 |
-3 | 0 |
-2 | 0 |
-1 | -4 |
0 | -6 |
1 | 0 |
The problem asks us to identify the y-intercept of a continuous function \(f(x)\) given by a set of coordinates in a table. A y-intercept occurs where the graph of the function crosses the vertical y-axis, which is equivalent to saying the input value \(x\) is equal to \(0\). The corresponding point on the graph is represented by the coordinate pair \((0, f(0))\).
We examine the provided table to identify the entry where the x-coordinate is zero:
\begin{itemize}
\item When \(x = -4\), \(f(x) = -10\)
\item When \(x = -3\), \(f(x) = 0\)
\item When \(x = -2\), \(f(x) = 0\)
\item When \(x = -1\), \(f(x) = -4\)
\item When \(x = 0\), \(f(x) = -6\)
\item When \(x = 1\), \(f(x) = 0\)
\end{itemize}
Looking at the row where \(x = 0\), we see that the function value \(f(0)\) is \(-6\). Therefore, the point on the function corresponding to the y-intercept is \((0, -6)\).
Comparing our result \((0, -6)\) with the given options:
\begin{itemize}
\item \((0, -6)\) - This matches our result.
\item \((-2, 0)\) - This is an x-intercept (since \(f(x) = 0\)).
\item \((-6, 0)\) - This is not on the function graph.
\item \((0, -2)\) - This is not in the table.
\end{itemize}
Thus, the correct y-intercept is \((0, -6)\).
In real life, a y-intercept often represents the starting value or the initial state of a system. For example, if you are tracking the height of a plant (y) over time in weeks (x), the y-intercept (at x = 0) tells you the height of the plant at the exact moment you started measuring it. Identifying these key points helps in understanding trends and behavior in many fields, from physics experiments to business growth projections.