Respuesta :
Answer:
The function f(x) has smallest minimum y-value. The smallest minimum y-value is -5.
Step-by-step explanation:
The given function is
[tex]f(x)=4\sin(2x-\pi)-1[/tex]
We know that the value of sinθ lies between -1 and 1.
[tex]-1\leq \sin(2x-\pi)\leq 1[/tex]
Multiply 4 on each side.
[tex]-4\leq 4\sin(2x-\pi)\leq 4[/tex]
Subtract 1 from each side.
[tex]-5\leq 4\sin(2x-\pi)-1\leq 3[/tex]
[tex]-5\leq f(x)\leq 3[/tex]
It means minimum value of f(x) is -5.
From the given table of g(x) it is noticed that the minimum y-value is -3 at x=2.
The given function is
[tex]h(x)=(x-2)^2+4[/tex] .... (1)
The vertex form of a parabola is
[tex]P(x)=a(x-h)^2+k[/tex] .... (2)
Where, (h,k) is vertex.
On comparing (1) and (2), we get
[tex]a=1,h=2,k=4[/tex]
The vertex is (2,4). Since a=1>0, therefore it is an upward parabola and the vertex of an upward parabola is the point of minima.
For the function h(x) the minimum value of y is 4 at x=2.
The minimum y-value of function f(x), g(x) and h(x) are -5,-3 and 4 respectively.
Therefore function f(x) has smallest minimum y-value. The smallest minimum y-value is -5.
The function that has the smallest minimum y-value is function f(x)
The functions are given as:
[tex]f(x) = 4 \sin(2x - \pi)-1[/tex]
[tex]h(x) = (x - 2)^2 + 4[/tex]
On the table of g(x), the minimum value y is -3
Function h(x) is a quadratic function, that has a vertex (minimum) at (2,4).
This means that the minimum y value of function h(x) is 4
Lastly, the minimum y value from the graph of function f(x) is at y = -5
So, we have:
y= -3 ------ function g(x)
y= 4 ------ function h(x)
y= -5 ------ function f(x)
-5 is the smallest of these values
Hence, the function that has the smallest minimum y-value is function f(x)
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