The two statements that are correct are f(0) is greater than g(0), and f(2) is greater than g(2).
What is the general equation of parabola?
The general equation of a parabola is given as,
[tex]y=a(x-h)^2+k[/tex]
where (h,k) is the coordinate of the vertex of the parabola.
The equation of the parabola with vertex (1,9) can be written as,
[tex]y=a(x-h)^2+k\\\\y=a(x-1)^2+9[/tex]
Now, the value of a can be found by putting one zero of the parabola,
[tex]y=a(4-1)^2+9\\\\0=a(3)^2+9\\\\-9=a(3)^2\\\\-9=a9\\\\a=-1[/tex]
Thus, the equation of the parabola can be written as y=-(x-1)²+9.
A.) The value of the two functions when the value of x is -2 can be written as,
[tex]f(x)=-(x-1)^2+9\\\\f(-2)=-(-2-1)^2+9\\\\f(-2)=0[/tex]
[tex]g(x) = -3x + 2\\\\g(-2) = -3(-2) + 2\\\\g(-2)=8[/tex]
Therefore, f(-2) is less than g(-2).
B.) The value of the two functions when the value of x is -1 can be written as,
[tex]f(x)=-(x-1)^2+9\\\\f(-1)=-(-1-1)^2+9\\\\f(-2)=5[/tex]
[tex]g(x) = -3x + 2\\\\g(-1) = -3(-1) + 2\\\\g(-1)=5[/tex]
Therefore, f(-1) is equal to g(-1).
C.) The value of the two functions when the value of x is 0 can be written as,
[tex]f(x)=-(x-1)^2+9\\\\f(0)=-(0-1)^2+9\\\\f(0)=8[/tex]
[tex]g(x) = -3x + 2\\\\g(0) = -3(0) + 2\\\\g(0)=2[/tex]
Therefore, f(0) is greater than g(0).
D.) The value of the two functions when the value of x is 1 can be written as,
[tex]f(x)=-(x-1)^2+9\\\\f(1)=-(1-1)^2+9\\\\f(1)=9[/tex]
[tex]g(x) = -3x + 2\\\\g(1) = -3(1) + 2\\\\g(1)=1[/tex]
Therefore, f(1) is greater than g(1).
E.) The value of the two functions when the value of x is 2 can be written as,
[tex]f(x)=-(x-1)^2+9\\\\f(2)=-(2-1)^2+9\\\\f(2)=8[/tex]
[tex]g(x) = -3x + 2\\\\g(2) = -3(2) + 2\\\\g(2)=-4[/tex]
Therefore, f(2) is greater than g(2).
Thus, the two statements that are correct are f(0) is greater than g(0), and f(2) is greater than g(2).
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