Question

I need help with my homework. I have the answer but I need to know if my answer is correct please help thank you in advance

Answer 1

Step-by-step explanation

Since we have the parabola:

`f(x)=6x^2-54`

We first need to obtain the roots in order to compute the x-intercepts:

`\mathrm{Add\: }54\mathrm{\: to\: both\: sides}`
`6x^2-54+54=0+54`

Simplify:

`6x^2=54`
`\mathrm{Divide\: both\: sides\: by\: }6`
`(6x^2)/(6)=(54)/(6)`

Simplify:

`x^2=9`

For x^2 = f(a) the solutions are x= √f(a) :

`x=√(9),\: x=-√(9)`

Computing the square root we get:

`\mathrm{X\: Intercepts}\colon\: \mleft(3,\: 0\mright),\: \mleft(-3,\: 0\mright)`

b) Computing the vertex:

`\mathrm{The\: vertex\: of\: an\: up-down\: facing\: parabola\: of\: the\: form}\: y=ax^2+bx+c\: \mathrm{is}\: x_v=-(b)/(2a)`
`\mathrm{The\: parabola\: params\: are\colon}`
`a=6,\: b=0,\: c=-54`
`x_v=-(b)/(2a)`
`x_v=-(0)/(2\cdot\:6)`
`\mathrm{Simplify}\colon`
`x_v=0`

Plugging in x_v to find the y_v value:

`y_v=6\cdot\: 0^2-54`

Simplify:

`y_v=-54`
`\mathrm{Therefore\: the\: parabola\: vertex\: is}\colon`
`\mleft(0,\: -54\mright)`