This tutorial explains how to determine the concavity of quadratic functions using the second derivative and leading coefficient.
Concavity of Quadratic Functions
The concavity of a function is determined by the sign of its second derivative. For a quadratic function of the form:
\[ f(x) = a x^{2} + b x + c, \quad a \neq 0 \]The first and second derivatives are:
\[ f'(x) = 2 a x + b \] \[ f''(x) = 2 a \]Since \( f''(x) \) is constant and depends solely on \( a \), the concavity of the parabola is determined by the sign of \( a \):
- If \( a > 0 \), then \( f''(x) > 0 \) and the graph is concave up.
- If \( a < 0 \), then \( f''(x) < 0 \) and the graph is concave down.
Worked Examples
Below are examples illustrating these cases with detailed solutions.
Example 1: Quadratic Function Concavity
Determine the concavity of the quadratic function:
\[ f(x) = (2 - x)(x - 3) + 3 \]Solution to Example 1
First, expand and rewrite \( f(x) \):
\[ f(x) = -x^{2} + 5x - 3 \]The leading coefficient \( a = -1 \) is negative, so the graph is concave down.
Example 2: Quadratic Function Concavity
Determine the concavity of the quadratic function:
\[ f(x) = -2(x - 1)(x - 2) + 3 x^{2} \]Solution to Example 2
Expand and simplify \( f(x) \):
\[ f(x) = x^{2} + 6x - 4 \]The leading coefficient \( a = 1 \) is positive, so the graph is concave up.
Exercises and Answers
Determine the concavity of each quadratic function below:
- \( f(x) = (2 - x)(4 - x) \)
- \( f(x) = -2(x - 3)^{2} - 5 \)
- \( f(x) = x(x + 3) - 2(x - 3)^{2} \)
Answers
- Concave up (Expanding gives \( x^2 - 6x + 8 \), where \( a = 1 > 0 \))
- Concave down (Expanding gives \( -2x^2 + 12x - 23 \), where \( a = -2 < 0 \))
- Concave down (Expanding gives \( -x^2 + 15x - 18 \), where \( a = -1 < 0 \))