Question

Do the points (−1,9) and (2,4) represent the vertex and a sample point, respectively, of y=−2(x+1)2+9? Explain

Answer 1

Vertex: (-1,9) is true

Sample Points: (2,4) is not true

Explanation:

Given

`Vertex: (-1,9)`

`Sample\ Points: (2,4)`

Required

Determine if the vertex and sample points exist for
`y= -2(x+1)\²+9`

Solving for the vertex:

`y= -2(x+1)\²+9`

Writing the given equation in the form:

`y = ax^2 + bx + c`

Expand the bracket

`y= -2(x+1)(x+1)+9`

Open Bracket

`y= -2(x+1)(x+1)+9`

`y = -2(x^2 + 2x + 1) + 9`

Open bracket

`y = -2x^2 - 4x -2 + 9`

`y = -2x^2 - 4x +7`

Solve for x using:

`x = (-b)/(2a)`

Where
`a = -2`;
`b = -4;`

So:

`x = (-(-4))/(2 * (-2))`

`x = (4)/(-4)`

`x = -1`

Substitute -1 for x in
`y = -2x^2 - 4x +7`

`y = -2(-1)^2 -4(-1) + 7`

Simplify all brackets

`y = -2(1) + 4 + 7`

`y = -2 + 4 + 7`

`y = 9`

Hence;

The vertex (x,y) is (-1,9)

This is true

Checking sample points:
`(2,4)`

In this case;
`x = 2` and
`y = 4`

Substitute
`x = 2` and
`y = 4` in
`y= -2(x+1)\²+9`

`4 = -2(2 + 1)^2 + 9`

`4 = -2(3)^2 + 9`

`4 = -2 * 9 + 9`

`4 = -18 + 9`

`4 \\eq -9`

Hence;

This sample points
`(2,4)` does not exist