Answer :
Test statement #1.
f(x) = -0.3(x - 5)² + 5
This is the equation of a parabola with vertex at (5,5).
Therefore the function is symmetric about x=5.
The statement "The axis of symmetry is x=5" is TRUE.
Test statement #2.
f(x) is defined for all real values of x.
The statement "The domain is {x | x is a real nuber} is TRUE.
Test statement #3.
As x -> -∞, f(x) -> -∞.
f(5) = -0.3*(5-5)^2 + 5 = 5
Therefore f(x) is creasing over (-∞, 5) is TRUE.
Test statement #4.
As x -> +∞, f(x) -> -∞.
Therefore the curve is concave downward., and it has no minimum.
The statement "The minimum is (5,5)" is False.
Test statement #5.
The maximum value of f(x) occurs at the vertex because the curve is concave downward.
The statement "The range is {y | y≥5}" is False.
Answer:
The first three statements are True. The last two statements are False.
f(x) = -0.3(x - 5)² + 5
This is the equation of a parabola with vertex at (5,5).
Therefore the function is symmetric about x=5.
The statement "The axis of symmetry is x=5" is TRUE.
Test statement #2.
f(x) is defined for all real values of x.
The statement "The domain is {x | x is a real nuber} is TRUE.
Test statement #3.
As x -> -∞, f(x) -> -∞.
f(5) = -0.3*(5-5)^2 + 5 = 5
Therefore f(x) is creasing over (-∞, 5) is TRUE.
Test statement #4.
As x -> +∞, f(x) -> -∞.
Therefore the curve is concave downward., and it has no minimum.
The statement "The minimum is (5,5)" is False.
Test statement #5.
The maximum value of f(x) occurs at the vertex because the curve is concave downward.
The statement "The range is {y | y≥5}" is False.
Answer:
The first three statements are True. The last two statements are False.
To solve the problem we must know about the Equation of a parabola and its function.
Equation of a parabola
y = a(x-h)2 + k
where,
(h, k) are the coordinates of the vertex of the parabola in form (x, y);
a defines how narrower is the parabola, and the "-" or "+" that the parabola will open up or down.
Function
A function assigns the value of each element of one set to the other specific element of another set.
For function [tex]f(x) = -0.3(x - 5)^2 + 5[/tex] , the options 1, 2, and 3 are correct.
Explanation
1. The axis of symmetry is [tex]x = 5[/tex].
This is the equation of a parabola with vertex at (5,5).
Thus, the function is symmetric about x=5.
Therefore, the statement is True.
2. The domain is [tex]\rm \{x\;|\; x\; is \;a \;real \;number\}[/tex]f(x) is defined for all real values of x.
Therefore, the statement is True.
3. The function is increasing over (–∞, 5)As x tends towards -∞, f(x) tends towards -∞.[tex]\bold{f(5) = -0.3\times(5-5)^2 + 5 = 5}[/tex]
Therefore, the statement is True.
4. The minimum is (5, 5).
The statement is False.
5. The range is {y| y ≥ 5}.
The maximum value of f(x) occurs at the vertex as the curve is concave downward.
Therefore, the statement is False.
For function [tex]f(x) = -0.3(x - 5)^2 + 5[/tex], all the options are true except statement 4 and 5.
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