- Calculate the \(f\)-intercept by evaluating \(f(0) = 84\), resulting in the point \((0, 84)\).
- Find the \(r\)-intercepts by setting \(f(r) = 0\) and solving \((r - 7)(r + 2)(r - 6) = 0\), yielding \(r = 7, -2, 6\).
- The \(f\)-intercept is \(\boxed{(0, 84)}\).
- The \(r\)-intercepts are \(\boxed{(7, 0), (-2, 0), (6, 0)}\).
Explanation
- Understanding the Problem
The function is given by \(f(r) = (r - 7)(r + 2)(r - 6)\). We need to find the \(f\)-intercept (where the curve crosses the vertical axis, at \(r = 0\)) and the \(r\)-intercepts (where the curve crosses the horizontal axis, at \(f(r) = 0\)). This is a cubic polynomial presented in factored form, which makes finding its roots quite straightforward using the zero-product property.
- Finding the f-intercept
To find the \(f\)-intercept, we evaluate the function at \(r = 0\):
\[f(0) = (0 - 7)(0 + 2)(0 - 6)\]
\[f(0) = (-7) \times (2) \times (-6)\]
\[f(0) = -14 \times (-6)\]
\[f(0) = 84\]
Thus, the \(f\)-intercept is at the coordinate \((0, 84)\).
- Finding the r-intercepts
To find the \(r\)-intercepts, we solve the equation \(f(r) = 0\):
\[(r - 7)(r + 2)(r - 6) = 0\]
By the zero-product property, the equation holds if any of its factors equal zero:
- \(r - 7 = 0 \implies r = 7\)
- \(r + 2 = 0 \implies r = -2\)
- \(r - 6 = 0 \implies r = 6\)
These values correspond to the points \((7, 0)\), \((-2, 0)\), and \((6, 0)\).
Examples
The concept of intercepts is fundamental in physics and engineering for understanding system states. For example, if you model the altitude of a projectile over time \(r\) with a function, the \(f\)-intercept tells you the initial height of the launch, while the \(r\)-intercepts (the roots) reveal the times when the projectile hits the ground. Understanding these points allows engineers to predict the trajectory and safety of mechanical systems effectively.