Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y = –3(x + 7)^2 – 10`?

Answer 1

`color(purple)("Thus the vertex is a maximum")`
`color(purple)("Vertex"->(x,y)=(-7,-10))`
`color(purple)("Axis of symmetry "->x=-7)`

Explanation:

This is the vertex form of equation type `y=ax^2+bx+c`

If you were to square the brackets and multiply by the -3 the `x^2` term would be `-3x^2`. As this is negative the graph is of general shape `nn`. `color(purple)("Thus the vertex is a maximum")`
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The coordinates of the vertex can be read directly from the given equation but you will need to 'tweak' it a bit.

Given: `y=-3(xcolor(red)(+7))^2color(green)(-10)`

`x_("vertex")=(-1)xxcolor(red)(+7) =-7`

`y_("vertex")=color(green)(-10)`

`color(purple)("Vertex"->(x,y)=(-7,-10))`
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The axis of symmetry passes through the vertex and is parallel to the x-axis (for a quadratic in x as is this one)

`color(purple)("Axis of symmetry "->x=-7)`