Sure, let's analyze the function [tex]\( f(x) = -0.3(x-5)^2 + 5 \)[/tex] and identify its key features step by step.
1. Axis of Symmetry:
The axis of symmetry of a quadratic function in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex] is given by the line [tex]\( x = h \)[/tex]. In this function, [tex]\( h = 5 \)[/tex]. Therefore, the axis of symmetry is [tex]\( x = 5 \)[/tex].
2. Domain:
The domain of a quadratic function, unless otherwise restricted, is all real numbers. Hence, for this function, the domain is [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex].
3. Intervals of Increase and Decrease:
- To find the intervals where the function is increasing or decreasing, we examine the vertex of the parabola.
- The function [tex]\( f(x) = -0.3(x-5)^2 + 5 \)[/tex] opens downwards because the coefficient of the quadratic term ([tex]\(-0.3\)[/tex]) is negative.
- The vertex of this parabola, which is at [tex]\( (5, 5) \)[/tex], is a maximum point.
- The function is increasing to the left of the vertex and decreasing to the right of the vertex.
- Therefore, the function is increasing over the interval [tex]\( (-\infty, 5) \)[/tex].
4. Maximum/Minimum Points:
- For the function [tex]\( f(x) = -0.3(x-5)^2 + 5 \)[/tex], the vertex is at [tex]\( (5, 5) \)[/tex].
- Since the parabola opens downward, this vertex is a maximum point.
- Hence, the correct statement is that the maximum is [tex]\( (5, 5) \)[/tex], not the minimum.
5. Range:
- Since the parabola opens downwards and the maximum value of the function is 5 (at [tex]\( x = 5 \)[/tex]), the range of the function includes all values [tex]\( y \)[/tex] such that [tex]\( y \leq 5 \)[/tex].
- Therefore, the range is [tex]\( \{ y \mid y \leq 5 \} \)[/tex].
Considering all the above points:
- 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].
- The maximum is [tex]\( (5, 5) \)[/tex].
- The range is [tex]\( \{ y \mid y \leq 5 \} \)[/tex].
Incorrect Statements:
- The minimum is [tex]\( (5, 5) \)[/tex] - this is incorrect - it is actually the maximum.
- The range is [tex]\( \{ y \mid y \geq 5 \} \)[/tex] - this is incorrect, as the correct range is [tex]\( \{ y \mid y \leq 5 \} \)[/tex].
Identified correct key features:
- 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].
