- Rewrite the function \(f(p) = p^2 - 8p - 5\) in the standard form \(ap^2 + bp + c\).
- Identify the coefficient \(a\) for the \(p^2\) term as \(1\).
- Identify the coefficient \(b\) for the \(p\) term as \(-8\).
- Identify the constant term \(c\) as \(-5\).
- The result is \(\boxed{a = 1, b = -8, c = -5}\).
Explanation
- Analyzing the Quadratic Function
The quadratic function is given as \(f(p) = p^2 - 8p - 5\). The general standard form of a quadratic function is \(f(p) = ap^2 + bp + c\), where \(a\), \(b\), and \(c\) are constants, and \(a
eq 0\). Our goal is to determine the specific values of \(a\), \(b\), and \(c\) by matching the terms of the given expression to this standard form.
- Identifying the Coefficients and Constant
To identify the coefficients, we can rewrite the function \(f(p) = p^2 - 8p - 5\) to clearly distinguish the coefficients for each term of \(p\):
- The term \(p^2\) can be written as \(1 \cdot p^2\), which indicates that \(a = 1\).
- The term \(-8p\) represents the linear term, where \(b = -8\).
- The constant term is \(-5\), so \(c = -5\).
Thus, comparing the given function to the standard form \(ap^2 + bp + c\), we find \(a = 1\), \(b = -8\), and \(c = -5\).
- Final Conclusion and Selection
Based on our identification, the coefficients are \(a = 1\), \(b = -8\), and \(c = -5\). Reviewing the given options:
- \(a = -1, b = -8, c = -5\)
- \(a = 1, b = -5, c = -8\)
- \(a = 1, b = -8, c = -5\)
- \(a = -1, b = -5, c = 8\)
The correct option is \(a = 1, b = -8, c = -5\), which corresponds to the third option. Therefore, the values are \(\boxed{a = 1, b = -8, c = -5}\).
Examples
In physics, quadratic functions often model the trajectory of a projectile. If a ball is thrown into the air, its height \(h\) at time \(t\) might be given by \(h(t) = -16t^2 + v_0t + h_0\). By identifying the coefficients \(a = -16\), \(b = v_0\), and \(c = h_0\), scientists can determine the initial velocity (\(v_0\)) and the starting height (\(h_0\)) to predict where and when the ball will land. Learning to identify these coefficients is a foundational step in modeling physical motion.