Verified Answer :

  • The function is \(f(x) = 2x - 6\).
  • The slope is \(m = 2\).
  • The y-intercept is \((0, -6)\).
  • To graph, plot the point \((0, -6)\), move up 2 units and right 1 unit to mark a second point at \((1, -4)\), then draw a straight line through these points.
  • Final answer: Slope \(m = 2\), y-intercept \(b = -6\).

Explanation

  1. Identify the Slope and y-intercept

The given function is a linear equation in the form \(f(x) = 2x - 6\). A linear function is typically expressed in the slope-intercept form, given by:
\[f(x) = mx + b\]
where \(m\) represents the slope of the line and \(b\) represents the y-intercept. By comparing \(f(x) = 2x - 6\) to the standard form \(f(x) = mx + b\), we can identify these parameters directly:

  • The coefficient of \(x\) is \(2\), so the slope is \(m = 2\).
  • The constant term is \(-6\), so the y-intercept is \(b = -6\).

This means the graph of the function crosses the y-axis at the point \((0, -6)\).

  1. Graphing the Function

To graph the function \(f(x) = 2x - 6\), we follow these steps:

  1. Plot the y-intercept: Since the y-intercept is \((0, -6)\), locate this point on the coordinate plane.
  2. Use the slope: The slope \(m = 2\) can be written as \(\frac{2}{1}\), which indicates a "rise over run" of \(\frac{2}{1}\). Starting from the y-intercept \((0, -6)\), move up 2 units and right 1 unit to reach the next point.
  3. Find the new point: Moving up 2 units from \(-6\) brings us to \(-4\), and moving right 1 unit from \(0\) brings us to \(1\). Thus, the second point is \((1, -4)\).
  4. Draw the line: Using a straightedge, draw a line passing through the points \((0, -6)\) and \((1, -4)\). This line represents the graph of the linear function \(f(x) = 2x - 6\).

Examples

Linear functions are fundamental in modeling real-world situations where one quantity changes at a constant rate relative to another. For example, if you are filling a pool that starts with 0 gallons of water and fills at a rate of 2 gallons per minute, the volume of water \(V\) after \(t\) minutes can be modeled by \(V(t) = 2t\). If the pool already had a small amount of water pumped out (like a leak) represented by \(-6\), the function \(V(t) = 2t - 6\) could model the water level over time. Understanding slopes and intercepts allows you to predict future values and analyze rates of change in fields ranging from economics to engineering.

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