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

Answer:

Vertex: (-1,9) is true

Sample Points: (2,4) is not true

Step-by-step 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:

$x = \frac{-b}{2a}$

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

So:

$x = \frac{-(-4)}{2 * (-2)}$

$x = \frac{4}{-4}$

$x = -1$

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 $y= -2(x+1)\²+9$

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

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

$4 = -2 * 9 + 9$

$4 = -18 + 9$

$4 \neq -9$

Hence;

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