Question

What is the axis of symmetry and vertex for the graph `y=-2x^2+4x+3`?

Answer 1

Axis of symmetry`" "->x-1`
`color(white)(.)`
Vertex`" "->(x,y)->(1,5)`

Explanation:

First consider the `-2x`. As this is negative the general shape of the graph is `nn`

The axis of symmetry will be parallel to the y-axis (normal to the x-axis) and pass through the vertex
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This next bit is a variant on the vertex form equation

Given:`" "y=-2x^2+4x+3" "`........................................(1)

Write as:`" "y=-2(x^2-4/2x)+3`

Consider the `-4/2 " from "-4/2x`

Apply this process:`" "(-1/2)xx(-4/2)=+1`

This value of `+1` is the value of `x_("vertex")`

`color(brown)("So "x=1 " is the axis if symmetry.")`
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Substitute `x=1` into equation (1) to find`" "y_("vertex")`

`=>y=-2(1)^2 +4(1)+3 = 5`

`color(brown)("Vertex" ->(x,y)->(1,5))`