Answer:
Vertex: (-1,9) is true
Sample Points: (2,4) is not true
Step-by-step explanation:
Given
[tex]Vertex: (-1,9)[/tex]
[tex]Sample\ Points: (2,4)[/tex]
Required
Determine if the vertex and sample points exist for [tex]y= -2(x+1)\²+9[/tex]
Solving for the vertex:
[tex]y= -2(x+1)\²+9[/tex]
Writing the given equation in the form:
[tex]y = ax^2 + bx + c[/tex]
Expand the bracket
[tex]y= -2(x+1)(x+1)+9[/tex]
Open Bracket
[tex]y= -2(x+1)(x+1)+9[/tex]
[tex]y = -2(x^2 + 2x + 1) + 9[/tex]
Open bracket
[tex]y = -2x^2 - 4x -2 + 9[/tex]
[tex]y = -2x^2 - 4x +7[/tex]
Solve for x using:
[tex]x = \frac{-b}{2a}[/tex]
Where [tex]a = -2[/tex]; [tex]b = -4;[/tex]
So:
[tex]x = \frac{-(-4)}{2 * (-2)}[/tex]
[tex]x = \frac{4}{-4}[/tex]
[tex]x = -1[/tex]
Substitute -1 for x in [tex]y = -2x^2 - 4x +7[/tex]
[tex]y = -2(-1)^2 -4(-1) + 7[/tex]
Simplify all brackets
[tex]y = -2(1) + 4 + 7[/tex]
[tex]y = -2 + 4 + 7[/tex]
[tex]y = 9[/tex]
Hence;
The vertex (x,y) is (-1,9)
This is true
Checking sample points: [tex](2,4)[/tex]
In this case; [tex]x = 2[/tex] and [tex]y = 4[/tex]
Substitute [tex]x = 2[/tex] and [tex]y = 4[/tex] in [tex]y= -2(x+1)\²+9[/tex]
[tex]4 = -2(2 + 1)^2 + 9[/tex]
[tex]4 = -2(3)^2 + 9[/tex]
[tex]4 = -2 * 9 + 9[/tex]
[tex]4 = -18 + 9[/tex]
[tex]4 \neq -9[/tex]
Hence;
This sample points [tex](2,4)[/tex] does not exist
