Graphical Analysis of Range of Quadratic Functions
The range of a function \( y = f(x) \) is the set of values \( y \) takes for all values of \( x \) within the domain of \( f \).
The graph of any quadratic function, of the form \( f(x) = ax^2 + bx + c \), can be written in vertex form as follows:
\[ f(x) = a(x - h)^2 + k \]where \( h = -\dfrac{b}{2a} \) and \( k = f(h) \).
The graph is either a parabola opening up when \( a > 0 \), or a parabola opening down when \( a < 0 \). Therefore:
- If \( a > 0 \), the graph of \( f \) has a minimum point at the vertex \( (h, k) \), and the range is \([k, +\infty)\).
- If \( a < 0 \), the graph of \( f \) has a maximum point at the vertex \( (h, k) \), and the range is \((-\infty, k]\).
Examples with Solutions
Example 1
Find the range of function \( f \) defined by:
\[ f(x) = -2x^2 + 4x + 2 \]View Solution
The vertex of the graph of \( f \) is at the point \( (h, k) \) where:
\[ h = -\dfrac{b}{2a} = -\dfrac{4}{2(-2)} = 1 \quad \text{and} \quad k = f(1) = 4 \]The leading coefficient \( a = -2 \) is negative, and therefore the graph of \( f \) has a maximum at the point \( (1, 4) \). The maximum value of \( f \) is 4. Hence, the range of \( f \) is given by the interval:
\[ (-\infty, 4] \]
Example 2
Find the range of function \( f \) defined by:
\[ f(x) = 2x^2 + 12x + 16 \]View Solution
The coordinates \( h \) and \( k \) of the vertex of the graph of \( f \) are given by:
\[ h = -\dfrac{b}{2a} = -\dfrac{12}{2(2)} = -3 \quad \text{and} \quad k = f(-3) = -2 \]The leading coefficient \( a = 2 \) is positive, and therefore the graph of \( f \) has a minimum point at \( (h, k) = (-3, -2) \). The range of \( f \) is given by the interval:
\[ [-2, +\infty) \]