$x$ | $f ( x )$ |
-2 | 15 |
-1 | 10 |
0 | 5 |
1 | 0 |
2 | -5 |
3 | -10 |
The y-intercept of a function \(f(x)\) is the point where the graph of the function intersects the y-axis. This occurs when the value of the independent variable, \(x\), is zero. Thus, a y-intercept is expressed as the coordinate pair \((0, f(0))\).
We examine the provided table of values for the function \(f(x)\):
| \(x\) | \(f(x)\) |
| :--- | :--- |
| \(-2\) | \(15\) |
| \(-1\) | \(10\) |
| \(0\) | \(5\) |
| \(1\) | \(0\) |
| \(2\) | \(-5\) |
| \(3\) | \(-10\) |
Locating the row where \(x = 0\), we see that the corresponding value for \(f(x)\) is \(5\). Therefore, the function evaluates to \(f(0) = 5\).
Since the y-intercept is the point \((0, f(0))\), and we have determined that \(f(0) = 5\), the y-intercept is \((0, 5)\). Comparing this to the given choices:
The point \((0, 5)\) is clearly the correct choice.
In real-world applications, identifying the y-intercept is crucial when analyzing initial conditions. For instance, if a company models its daily profit \(P\) as a function of time \(t\) in days, the y-intercept \((0, P(0))\) represents the profit (or loss) at the very start of the observation period (\(t=0\)). Similarly, if you are tracking the height of a plant over time, the y-intercept tells you exactly how tall the plant was when you first started measuring it. By looking at a table or graph of data, you can easily determine these starting values to make better predictions.