(03.02 MC)Two students, Carlos and Sophie, factored the trinomial 3y2+24y+453y2+24y+45. Sophie factored it as 3(y+3)(y+5) and Carlos factored it as (3y+9)(y+5). Indicate which student factored the trinomial completely and which student did not, and explain why. (10 points)Three functions are given below: f(x), g(x), and h(x). Explain how to find the axis of symmetry for each function, and rank the functions based on their axis of symmetry (from smallest to largest).f(x) g(x) h(x)f(x) = -4(x − 8)2 + 3 g(x) = 3x2 + 12x + 15 graph of negative 1 times the quantity of x minus 3 squared, plus 2(10 points)(03.06 LC)The function f(x) = -(x + 1)2 + 25 has been rewritten using the completing-the-square method. The function g(x)=3x2+30x+93x2+30x+9Which vertex for each function has a minimum or a maximum? What is the vertex for each function? Explain your reasoning for each function.(10 points)

1. We can determine which of the students factored the trinomial completely by expanding the factored form for each.

Sophie: [tex]3(y+3)(y+5)=3( y^{2} +8y+15)=3 y^{2} +24y+45[/tex]
Carlos: [tex](3y+9)(y+5)=3 y^{2}+24y+45 [/tex]

As we can see, both answers of the students equaled to the original trinomial when expanded. Therefore, both students factored the trinomial correctly. However, one student had a more complete factorization than the other, and that student is Sophie since she still isolated the common term in the first factor which Carlos failed to do.

2. The graphs given take the form of the parabola. Therefore, to find the axis of symmetry, we just need to find the x component of the vertex.

The equations follow the form [tex]y=a(x-h)^{2}+k[/tex] where h is the x component of the vertex.

By examining the equations below, we can find the axis of symmetry for each:
[tex]f(x)=-4(x-8)^{2} +3[/tex]
[tex]g(x)=3x^{2}+12x+15=3(x+2)^{2} +3[/tex]
[tex]h(x)=-1(x-3)^{2} +2[/tex]

The axis of symmetry for f(x), g(x), and h(x), respectively, are as follows:
[tex]x=8[/tex]; [tex]x=-2[/tex]; [tex]x=3[/tex]

Ranking the three functions from smallest to largest based on their axis of symmetry we'll get: g(x), h(x), and f(x).

3. Again, we look at the form of the parabola [tex]y=a(x-h)^{2}+k[/tex] to determine which function will have the minimum or a maximum. Let's first express both functions to this form by completing the square if necessary.

[tex]f(x)=-(x+1)^{2}+25[/tex]
[tex]g(x)=3x^{2}+30x+9=3(x+5)^{2}-66[/tex]

Through the coefficient a, we can tell if the parabola is facing up or down. If the value of a is positive, then the parabola is facing up; otherwise it's facing down if a is negative.

By examining the functions we can tell that f(x) is facing down while g(x) is facing up. Therefore, we know that f(x) will have a maximum value and g(x) will have a minimum value. We know that these values would be the vertex of the two functions.

The vertex follows the form (h,k) based on the equation of the parabola given above. Thus, the vertex of f(x) would be (-1,25) while the vertex of g(x) would be (-5,-66).