Question

How do you find the vertex and intercepts for `y = -4(x + 3)^2 - 6`?

Answer 1

`x_("intercepts") -> "none"`
`y_("intercepts") -> -42`

`"vertex "->(x,y)->(-3,-6)`

Explanation:

Given: `" "y=-4(x+3)^2-6`

This equation is quadratic in Vertex Form.

Notice that if you expand the brackets you would have `-4x^2`
As the coefficient of `x^2` is negative it means that you have a graph of the form `nn`

'~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find the vertex")`

Consider the content of the brackets. You have +3.

Multiply this by negative 1 and you have the vertex x-value

`color(blue)(x_("vertex")=(-1)xx3=-3)`
Substitute -3 for `x` in the equation and you have

`y_("vertex")=-4(-3+3)^2-6`

`color(blue)(y_("vertex")= 0-6" "=" "-6)`

`color(brown)( "vertex " ->" "(x,y)" "->" "(-3,-6)" " )`(below the x-axis)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("To find the x-intercepts")`

And is of shape `nn` then the curve does not cross the x=axis
so there is no intercepts on the x-axis

`color(blue)("None")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find the y-intercept")`

The y-axis intercept is when `x=0`

`y_("intercept") =-4(0+3)^2-6`

`color(blue)(y_("intercept") =-42)`