Let's analyze the function [tex]\( f(x) = -0.3(x-5)^2 + 5 \)[/tex] to identify its key features.
1. Axis of Symmetry:
The function is in the vertex form [tex]\( y = a(x-h)^2 + k \)[/tex], which allows us to directly identify the axis of symmetry. In this function, [tex]\( h = 5 \)[/tex]. Thus, the axis of symmetry is [tex]\( x = 5 \)[/tex].
- Correct: The axis of symmetry is [tex]\( x = 5 \)[/tex].
2. Domain:
A quadratic function of the form [tex]\( y = a(x-h)^2 + k \)[/tex] typically has a domain of all real numbers. There are no restrictions on the input values [tex]\( x \)[/tex] for a polynomial function.
- Correct: The domain is [tex]\( \{x \mid x \text{ is a real number} \} \)[/tex].
3. Increasing and Decreasing Intervals:
Because [tex]\( a = -0.3 \)[/tex], which is negative, the parabola opens downward. This implies that the function is increasing on the interval to the left of the vertex and decreasing on the interval to the right of the vertex. Specifically, the function is increasing over [tex]\( (-\infty, 5) \)[/tex] and decreasing over [tex]\( (5, \infty) \)[/tex].
- Correct: The function is increasing over [tex]\( (-\infty, 5) \)[/tex].
4. Maximum or Minimum Value:
For a downward-opening parabola, the vertex represents the maximum point on the graph. The vertex can be directly extracted from the function form [tex]\( y = -0.3(x-5)^2 + 5 \)[/tex]. Here, the vertex is at [tex]\( (5, 5) \)[/tex]. Since the parabola opens downward, this is a maximum value, not a minimum.
- Incorrect: The claim that the minimum is [tex]\( (5, 5) \)[/tex] is false. The correct statement would be that the maximum is [tex]\( (5, 5) \)[/tex].
5. Range:
Because the function reaches its maximum at [tex]\( y = 5 \)[/tex] and the parabola opens downward, all the output values of [tex]\( y \)[/tex] will be less than or equal to 5. Hence, the range is [tex]\( \{y \mid y \leq 5\} \)[/tex].
- Incorrect: The range is not [tex]\( \{y \mid y \geq 5\} \)[/tex]; it should be [tex]\( \{y \mid y \leq 5\} \)[/tex].
Summarizing all correct statements:
- The axis of symmetry is [tex]\( x = 5 \)[/tex].
- The domain is [tex]\( \{x \mid x \text{ is a real number} \} \)[/tex].
- The function is increasing over [tex]\( (-\infty, 5) \)[/tex].
