Question

What is the vertex, axis of symmetry, maximum or minimum value, and the range of the parabola `f(x) = 3x^2 - 4x -2`?

Answer 1

Minimum
`x_("intercepts") ~~ 1.721 and 0.387` to 3 decimal places
`y_("intercept")=-2`

Axis of symmetry `x=2/3`

Vertex `->(x,y)=(2/3,-10/3)`

Explanation:

The term `3x^2` is positive so the graph is of shape type `uu` thus a `color(blue)("minimum")`

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Write as `3(x^2-4/3x)-2`

`color(blue)("So the axis of symmetry is "x=(-1/2)xx-4/3 = +2/3)`

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Thus `x_("vertex")=2/3`

By substitution `y_("vertex")=3(2/3)^2-4(2/3)-2 = -3.33bar(3)=-10/3`

`color(blue)("Vertex "->(x,y)=(2/3,-10/3)`
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Read directly from `f(x)=3x^2-4x-2`

`color(blue)(y_("intercept")=-2)`

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To find the roots by completing the square we have

`y=3(x-4/(3xx2))^2+k-2`
`=> 3(-4/6)^2+k=0 => k=-16/12 = -4/3` giving

`y=3(x-2/3)^2-4/3-2`

`y=3(x-2/3)^2-10/3`
This confirms the vertex as `+2/3 and -10/3`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Set `y=0`

`3(x-2/3)^2=10/3`

`x-2/3=+-sqrt(10/9)`

`x=2/3+-sqrt(10)/3`

`x~~ 1.721 and 0.387` to 3 decimal places