Question

How do you find the vertex of `f(x)= -3x^2-6x-7`?

Answer 1

Vertex`->(x,y)->(-1,-4)`

Solved using Calculus and by not using Calculus

Explanation:

As you are using `f(x)` I am assuming you are at a higher level of Mathematics.
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`color(brown)("Using Calculus with shortcuts")`

`color(blue)("To find "x_("vertex")`

Given: `f(x)=-3x^2-6x-7`..............(1)

`f'(x)=-6x-6`

Equating to zero gives `color(blue)(x_("vertex")=-1)`
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`color(blue)("To find "y_("vertex")`

Substitute `x=-1` into equation (1) giving

`y_("vertex")= -3(-1)^2-6(-1)-7`

`color(blue)(y_("vertex")= -4)`
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Vertex`->(x,y)->(-1,-4)`

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This is a maximum as `-3x^2` being negative is indicative of an inverted U shape. Also the 2nd differential is negative which is also indicative of a maximum
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`color(brown)("NOT using Calculus")`

`color(blue)("I am going to show you a really cool trick!")`

Write as:`y= (-3x^2-6x)-7`

Factor the -3 out

`y= -3(x^2+2x)-7`............................(1)

Now consider the `+2" from "2x` inside the bracket

Multiply this by `(-1/2)`

`(-1/2)xx(+2)=-1`
This is the value you are after for `x` so:

`color(blue)(x_("vertex")=-1)`

Substitute this x value into the original equation to find the value of `y_("vertex")`