Write a polynomial function with rational coefficients so that P(x)=0 has the given root.
1 answer:
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The given function is:
The parents functions of g(x) will be:
The domain of g(x) and its parent function is the same i.e. Set of all Real numbers except 0.
The range of g(x) and its parent function is the same i.e. set of all real numbers except 0.
g(x) and its parent function only decrease. They do not increase over any interval. However, the interval in which they decrease is the same for both.
So, the correct answers are:
The domain of g(x) is the same as the domain of the parent function.
<span>The range is the same as the range of the parent function.
</span><span>The function g(x) decreases over the same x-values as the parent function.</span>
The parents functions of g(x) will be:
The domain of g(x) and its parent function is the same i.e. Set of all Real numbers except 0.
The range of g(x) and its parent function is the same i.e. set of all real numbers except 0.
g(x) and its parent function only decrease. They do not increase over any interval. However, the interval in which they decrease is the same for both.
So, the correct answers are:
The domain of g(x) is the same as the domain of the parent function.
<span>The range is the same as the range of the parent function.
</span><span>The function g(x) decreases over the same x-values as the parent function.</span>
A parabolic function's key characteristic is either having 2 x-intercepts or 2 y-intercepts. That is the reason why the standard form of parabolic functions are:
(x-h)^2 = +/- 4a(y-k) or (y-k)^2 = +/- 4a(x-h), where
(h,k) is the coordinates of the vertex
4a is the lactus rectum
a is the distance from the focus to the vertex
This is also called vertex form because the vertex (h,k) is grouped according to their variable.
Since we don't know any of those parameters, we'll just have to graph the data points given as shown in the picture. From this data alone, we can see that the parabola has two x-intercepts, x=-4 and x=-2. Since it has 2 roots, the parabola is a quadratic equation. Its equation should be
y = (x+4)(x+2)
Expanding the right side
y = x²+4x+2x+8
y = x²+6x+8
Rearrange the equation such that all x terms are on one side of the equation
x²+6x+___=y-8+___
The blank is designated for the missing terms to complete the square. Through completing the squares method, you can express the left side of the equation into (x-h)² form. This is done by taking the middle term, dividing it by two, and squaring it. So, (6/2)²=9. Therefore, you put 9 to the 2 blanks. The equation is unchanged because you add 9 to both sides of the equation.
The final equation is
x²+6x+9=y-8+9
(x+3)²=y+1
(x-h)^2 = +/- 4a(y-k) or (y-k)^2 = +/- 4a(x-h), where
(h,k) is the coordinates of the vertex
4a is the lactus rectum
a is the distance from the focus to the vertex
This is also called vertex form because the vertex (h,k) is grouped according to their variable.
Since we don't know any of those parameters, we'll just have to graph the data points given as shown in the picture. From this data alone, we can see that the parabola has two x-intercepts, x=-4 and x=-2. Since it has 2 roots, the parabola is a quadratic equation. Its equation should be
y = (x+4)(x+2)
Expanding the right side
y = x²+4x+2x+8
y = x²+6x+8
Rearrange the equation such that all x terms are on one side of the equation
x²+6x+___=y-8+___
The blank is designated for the missing terms to complete the square. Through completing the squares method, you can express the left side of the equation into (x-h)² form. This is done by taking the middle term, dividing it by two, and squaring it. So, (6/2)²=9. Therefore, you put 9 to the 2 blanks. The equation is unchanged because you add 9 to both sides of the equation.
The final equation is
x²+6x+9=y-8+9
(x+3)²=y+1