Example 5: Graph y = x^2 + 2x + 3

Find the vertex and the axis of symmetry. Sketch these in.
• Find the x-intercept by plugging in 0 for y.
• Find the y-intercept by plugging in 0 for x.
• Reflect your points across the axis of symmetry and connect your dots with a smooth
U-shaped (not V-shaped) curve.

for now, I'm just gonna type my work and figure out what to do next

find the line of symmetry -
1. a = 1, b = 2, c = 3

use this to find the vertex
1. x = (b/2a)
2. x = -(2) / 2(1) = -1

since we know that the along the x axis at -1 will be the vertex we replace x with 1 in the original formula
1. x=1
2. y = x^2 -2x - 8
3. y = 1^2 + -2 * 1 - 8 = 1- 2 - 8 = -9
4. y = -9

the vertex is (1, -9)

since the vertex is -1,-9 we know that x=1 is the axis of symmetry

finding the y-intercept is the easiest to start with because we just replace x with 0

x = 0 | y = x^2 -2x - 8

y = 0 - 8 = -8

y-intercept = (0,-8)

so so To find the x-intercepts, you can set y equal to zero and solve for x:

y = 0 | x = (-b ± sqrt(b^2 - 4ac)) / 2a
1. x = (-(-2) ± sqrt((-2)^2 - 4(1)(-8))) / 2(1)
  
  x = (2 ± sqrt(4 + 32)) / 2
  
  x = (2 ± sqrt(36)) / 2
  
  x = (2 ± 6) / 2
  
  x = 8 / 2 or x = -4/ 2
  
  x = 4 or x = -2
  sooooo (-2,0) & (4,0)

so since we know 3 y axis points on the graph and the axis of symmetry we can get another point without doing much work
1. symmetry line = x = 1,
2. calc'd x-intercept 0,-8
  1. the symmetry line is 1 and the known point is 0 since 1-0 = 1 we can add that to the x coordinate of y and keep the same y coordinate to get the mirrored point making another point on the graph (2,-8)
3. since we need one more point for the graph we can choose say x=3, | x^2 -2x - 8
  1. y = 3^2 -3*2 - 8 = -5
    1. soooo the new point is (3,-5) if we mirror that along 1,-9 we get (-1, -5 ) because 3 is 2 more than 1, and 2 less than 1 is -1. we also keep the same y coordinate
4. so all points are:
  1. (1, -9)
  2. (0,-8)
  3. (2, -8)
  4. (3,-5)
  5. (-1,-5)