Questions: Graph the equation y=-x^2+2x+8 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

Transcript text: Graph the equation $y=-x^{2}+2 x+8$ on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

Solution

Solution Steps

Step 1: Find the roots

To find the roots, set y = 0 and solve for x.

0 = -x² + 2x + 8 0 = x² - 2x - 8 0 = (x - 4)(x + 2) x = 4 or x = -2

The roots are x = 4 and x = -2. This gives us two points: (4, 0) and (-2, 0).

Step 2: Find the vertex

The x-coordinate of the vertex is given by x = -b/2a. In this case, a = -1 and b = 2.

x = -2 / (2 * -1) x = -2 / -2 x = 1

Now substitute x = 1 back into the original equation to find the y-coordinate of the vertex:

y = -(1)² + 2(1) + 8 y = -1 + 2 + 8 y = 9

The vertex is (1, 9).

Step 3: Find two additional points

We already have three points: (4, 0), (-2, 0), and (1, 9). We need two more. We can choose any x-values. Let's pick x = 0 and x = 2.

For x = 0: y = -(0)² + 2(0) + 8 y = 8 Point: (0, 8)

For x = 2: y = -(2)² + 2(2) + 8 y = -4 + 4 + 8 y = 8 Point: (2, 8)

Final Answer:

The five points are (-2, 0), (0, 8), (1, 9), (2, 8), and (4, 0). Plot these points on the graph and connect them with a smooth curve to complete the parabola.

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