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Answer with Step-by-step explanation:
We are given that function f(x) which is quadratic function.
x -intercept of function f(x) at (-1,0) and (-3,0)
x-Intercept of f means zeroes of f
x=-1 and x=-3
Range of f =[-4,)
g(x)=
Therefore, x-intercept of g(x) at (-1,0) and (-3,0).
Substitute x=-2
By comparing with the equation of parabola
Where vertex=(h,k)
We get vertex of g(x)=(-2,-2)
Range of g(x)=[-2,)
Zeroes of f and g are same .
But range of f and g are different.
Range of f contains -3 and -4 but range of g does not contain -3 and -4.
f and g are both quadratic functions.
There is a multiple zero at 0 (which means that it touches there), and there are single zeros at -2 and 2 (which means that they cross). There is also 2 imaginary zeros at i and -i.
You can find this by factoring. Start by pulling out the greatest common factor, which in this case is -x^2.
-x^6 + 3x^4 + 4x^2
-x^2(x^4 - 3x^2 - 4)
Now we can factor the inside of the parenthesis. You do this by finding factors of the last number that add up to the middle number.
-x^2(x^4 - 3x^2 - 4)
-x^2(x^2 - 4)(x^2 + 1)
Now we can use the factors of two perfect squares rule to factor the middle parenthesis.
-x^2(x^2 - 4)(x^2 + 1)
-x^2(x - 2)(x + 2)(x^2 + 1)
We would also want to split the term in the front.
-x^2(x - 2)(x + 2)(x^2 + 1)
(x)(-x)(x - 2)(x + 2)(x^2 + 1)
Now we would set each portion equal to 0 and solve.
First root
x = 0 ---> no work needed
Second root
-x = 0 ---> divide by -1
x = 0
Third root
x - 2 = 0
x = 2
Forth root
x + 2 = 0
x = -2
Fifth and Sixth roots
x^2 + 1 = 0
x^2 = -1
x = +/-
x = +/- i