Graph the equation $$ y=x^{2}-2 x-8 $$ on the accompanying including the roots and the vertex.

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We want to graph the quadratic

y = x² – 2x – 8.

Step 1. Find the Roots:
To find the x-intercepts, set y = 0:
  x² – 2x – 8 = 0.
Factor the quadratic:
  (x – 4)(x + 2) = 0.
So, the roots are:
  x – 4 = 0 ⟹ x = 4,
  x + 2 = 0 ⟹ x = –2.
These give the points (4, 0) and (–2, 0).

Step 2. Find the Vertex:
For a quadratic in the form y = ax² + bx + c, the x-coordinate of the vertex is given by
  x = –b/(2a).
Here, a = 1 and b = –2:
  x = –(–2)/(2·1) = 2/2 = 1.
Now substitute x = 1 into the equation to get y:
  y = (1)² – 2(1) – 8 = 1 – 2 – 8 = –9.
Thus, the vertex is at (1, –9).

Step 3. Find the y-intercept:
Set x = 0:
  y = 0² – 2·0 – 8 = –8.
So, the y-intercept is (0, –8).

Step 4. Graphing:
1. Plot the vertex at (1, –9). This is the lowest point on the parabola (since a > 0, the parabola opens upward).
2. Plot the roots (–2, 0) and (4, 0).
3. Plot the y-intercept (0, –8).
4. Draw a smooth curve through these points forming a “U” shape. The parabola is symmetric about the vertical line x = 1.

Here is a rough sketch of the graph:

y
         ↑
       5 |                   
       4 |                    
       3 |                    
       2 |                    
       1 |                    
   y=0 —+————•————•————•————→ x
         |   (–2,0)  (4,0)
      -1 |
      -2 |
      -3 |
      -4 |
      -5 |
      -6 |
      -7 |
      -8 |         • (0,–8)
      -9 |            • (1,–9)
    -10 |
         |
         +--------------------------------
               -2  -1   0   1   2   3   4

Remember, the parabola is symmetric about the vertical line line x = 1. You can plot additional points on both sides of the vertex if needed for accuracy.

That is the graph of y = x² – 2x – 8, complete with its roots (–2, 0) and (4, 0) and vertex (1, –9).
To graph the equation $$ y = x^{2} - 2x - 8 $$: 1. **Find the Roots**: Solve $$ x^{2} - 2x - 8 = 0 $$ to get $$ x = 4 $$ and $$ x = -2 $$. Plot points (4,0) and (-2,0). 2. **Find the Vertex**: Use $$ x = -\frac{b}{2a} = 1 $$. Substitute into the equation to find $$ y = -9 $$. Plot vertex at (1, -9). 3. **Find the y-intercept**: Set $$ x = 0 $$ to get $$ y = -8 $$. Plot point (0, -8). 4. **Graph the Parabola**: Draw a symmetric "U" shape with the vertex at (1, -9), passing through the roots and y-intercept. Here's a rough sketch: ``` y ↑ 5 | 4 | 3 | 2 | 1 | 0 —+————•————•————•————→ x | (-2,0) (4,0) -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | • (0,-8) -9 | • (1,-9) -10 | | +-------------------------------- -2 -1 0 1 2 3 4 ``` The parabola opens upward, symmetric about the line $$ x = 1 $$.

Graph the equation \( y=x^{2}-2 x-8 \) on the accompanying including the roots and the vertex.

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