- Define the function \(g(x) = 3(14x - 15) = 42x - 45\).
- Define the new function \(y = g(x) - 2 = 42x - 47\).
- Calculate the y-intercept by setting \(x = 0\): \(y(0) = 42(0) - 47 = -47\).
- The final answer is \(\boxed{-47}\), which corresponds to option A.
Explanation
- Understanding the Y-intercept
The function \(g(x)\) is defined as \(g(x) = 3(14x - 15)\). We are interested in the graph of a new function \(y = g(x) - 2\). To find the y-intercept of any function \(y = f(x)\) in the xy-plane, we look for the point where the graph crosses the y-axis, which occurs when the x-coordinate is zero. The y-coordinate of this point is found by calculating \(y(0)\).
- Defining the New Function
We are given the definition of \(g(x)\) as \(g(x) = 3(14x - 15)\). First, let's simplify this expression:
\[g(x) = 3 \times 14x - 3 \times 15 = 42x - 45.\]
Now, define the new function \(y = g(x) - 2\). Substituting the simplified expression for \(g(x)\):
\[y = (42x - 45) - 2.\]
Combining the constant terms:
\[y = 42x - 47.\]
- Calculating the Y-coordinate of the Y-intercept
To find the y-coordinate of the y-intercept, we substitute \(x = 0\) into our new function equation \(y = 42x - 47\):
\[y(0) = 42(0) - 47.\]
Calculating this:
\[y(0) = 0 - 47 = -47.\]
Therefore, the y-coordinate of the y-intercept of the graph of \(y = g(x) - 2\) is \(-47\). This corresponds to option A.
Examples
In a business scenario, imagine you are modeling your company's profit based on the number of items sold, where \(g(x)\) represents a base profit function and you subtract a fixed maintenance cost (represented by the -2). If you need to determine your starting financial position (the y-intercept, where no items have been sold, so \(x=0\)), you would apply this exact calculation to determine if your baseline is positive or negative. Understanding this helps in financial forecasting and determining break-even points.