Verified Answer :
- Identify the function as a linear equation in slope-intercept form \(y = mx + b\), where \(m = \frac{2}{3}\) and \(b = -2\).
- Note that the y-intercept \((0, -2)\) is the point where the line crosses the y-axis.
- Use the slope \(m = \frac{2}{3}\) to find another point on the line: for every 3 units to the right, move 2 units up from \((0, -2)\) to reach \((3, 0)\).
- Conclude that the correct graph must be a straight line passing through \((0, -2)\) and \((3, 0)\).
Explanation
- Analyzing the Function
The function provided is \(y = \frac{2}{3}x - 2\). This is written in the slope-intercept form, which is generally expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
Comparing our function to the general form:
- The y-intercept is \(b = -2\). This means the graph of the function crosses the y-axis at the point \((0, -2)\).
- The slope is \(m = \frac{2}{3}\). This represents the steepness and direction of the line. Since the slope is positive, the line rises as it moves from left to right. A slope of \(\frac{2}{3}\) means that for every 3 units you move horizontally to the right, you must move 2 units vertically upward.
- Finding Points on the Line
To accurately identify the graph, we can calculate at least one more point. Using the slope \(m = \frac{2}{3}\) starting from the y-intercept \((0, -2)\):
- Start at \((0, -2)\).
- Move right by 3 units: \(0 + 3 = 3\).
- Move up by 2 units: \(-2 + 2 = 0\).
Thus, the point \((3, 0)\) should also lie on the line. We can verify this by substituting \(x = 3\) into the original equation:
\(y = \frac{2}{3}(3) - 2 = 2 - 2 = 0\).
This confirms the point \((3, 0)\) is on the line. Any graph representing this function must pass through the points \((0, -2)\) and \((3, 0)\).
- Visual Characteristics and Conclusion
The graph of the function \(y = \frac{2}{3}x - 2\) is a straight line that intersects the y-axis at \((0, -2)\) and the x-axis at \((3, 0)\). Because the slope is positive, the line will be directed upwards as it moves to the right. To identify the correct choice from the provided options, look for the graph that crosses the y-axis at \(-2\) and passes through the point where \(x=3\) on the x-axis.
Examples
Understanding how to interpret the slope and intercept is crucial for many real-life applications. For example, if a taxi company charges a flat fee of \(\$2\) for starting the trip and then \(\$0.67\) (approx. \(\frac{2}{3}\)) per kilometer, the cost \(y\) for traveling \(x\) kilometers is given by \(y = \frac{2}{3}x + 2\). Recognizing this as a linear function allows you to quickly estimate costs for different distances, helping you manage your budget effectively. In this specific problem, we look at the graph where the y-intercept is \(-2\) instead of \(+2\), shifting the starting cost perspective to a different baseline.