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Example 4 Graph f(x) = x^2 -2x - 8

Steps to Graph ax^2 + bx + c

  • 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.

Graph f(x) = x^2 -2x - 8 

    the axis of symmetry is -9

    image.png

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

    1. fubdfind the line of symetrymsymmetry -(b/2a)=x
    1. a = 1, b = -2, c = -8
    use this to find the vertex
      x = (b/2a) 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
        x=1  
        y = x^2 -2x - 8 y = 1^2 + -2 * 1 - 8=8 = 1- 2 - 8 = -9 
        y = y-9 the vertex is x,(1, -99)
        findsince the vertex is -1,-9 we know that x=1 is the axis of symetrysymmetry finding the axis of symmetryy-intercept is the easiest to start with because we just replace x with 0 x = 0 | y = x^2 -92x - 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: x^

        y = 0 | x = (-b ± sqrt(b^2 - 2x4ac)) / 2a

          x = (-(-2) ± sqrt((-2)^2 - 84(1)(-8))) / 2(1)

          x = 0
          (x2 -± 4)(xsqrt(4 + 2)32)) / 2

          x = 0
          (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 
            symmetry line = x = 1,  calc'd x-intercept 0,-8
              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)  since we need one more point for the graph we can choose say x=3, | x^2 -2x - 8
                y = 3^2 -3*2 - 8 = -5
                  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 so all points are:
                    (1, -9) (0,-8) (2, -8) (3,-5) (-1,-5)

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