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How Do You Graph a Quadratic Function?

Graph y = x2 + 12x + 32.

Summary

  1. The axis of symmetry is the dashed 'imaginary' line that splits a parabola in half
  2. Our equation y=x2+12x+32 has the form y=ax2+bx+c
  3. So in our equation y=x2+12x+32, 'b' is 12 and 'a' is 1
  4. The vertex is the highest or lowest point on the graph of a parabola
  5. The axis of symmetry, x = -6, gives us the x-value of the vertex
  6. We plugged -8, -7, -6, -5, and -4 in for 'x' in our equation to find ordered pairs for graphing

Notes

    1. The axis of symmetry is an invisible line that divides a graphed line into identical halves
    2. 'a' is the coefficient in front of the x2 term in a quadratic function
    3. 'b' is the coefficient in front of the x term
    4. So 'b' is 12 and 'a' is 1 in our equation y=x2+12x+32
    1. Our equation y=x2+12x+32 has the form y=ax2+bx+c
    2. So 'b' is 12 and 'a' is 1 in our equation y=x2+12x+32
    3. x = -b/2a = -12/2(1) = -12/2 = -6
    4. So the vertical line through 'x=-6' will divide our graphed parabola into identical halves
    1. The vertex is the highest or lowest point on the graph of a parabola
    2. The vertex is where the axis of symmetry intersect the graphed line y=x2+12x+32
    3. This means we can plug 'x=-6' into our equation to get the y-value of the vertex
    1. The vertex is where the axis of symmetry intersects the graph of the parabola y=x2+12x+32
    2. This means we can plug 'x=-6' into our equation to get the y-value of the vertex
    3. y=x2+12x+32 = (-6)2+12(-6)+32 = 36-72+32 = -4
    1. We can organize the x-values we want to plug into y=x212x+32 using a table
    2. We know that parabolas are symmetric and that x = -6 at the vertex
    3. So we can choose values around x = -6 to find some points on our parabola
    4. Plugging 'x' into our original equation will let us solve for 'y'
    5. (-8)2+12(-8)+32 = 64-96+32 = 0
    6. (-7)2+12(-7)+32 = 49-84+32 = -3
    7. (-6)2+12(-6)+32 = 36-72+32 = -4
    8. (-5)2+12(-5)+32 = 25-60+32 = -3
    9. (-4)2+12(-4)+32 = 16-48+32 = 0
    1. Our ordered pairs to graph are:
    2. (-8,0)
    3. (-7,-3)
    4. (-6,-4)
    5. (-5,-3)
    6. (-4,0)
    7. Graphing the points and connecting them will give us a parabola
    1. After graphing the points and connecting them with a line, check the axis of symmetry
    2. A vertical line drawn through 'x=-6' should cut the parabola into equal halves
    3. Also, this vertical line should intersect the parabola at the vertex, (-6,-4) -- it does!