Quadratic Functions

Polynomial Function:  f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0
Constant Function:  f(x) = a
Linear Function:  f(x) = mx + b
Quadratic Function:  f(x) = ax2 + bx + c
Cubic Function:  f(x) = ax3 + bx2 + cx + d
Quartic Function:  f(x) = ax4 + bx3 + cx2 + dx + e
Quintic Function:  f(x) = ax5 + bx4 + cx3 + dx2 + ex + f

To graph y = ax2 + bx + c, use the following information:

  1. The equation of the axis of symmetry is x = -b

    2.                                                                   2a
  2. The x-value at the vertex of the parabola is x = -b

    3.                                                                        2a
  3. If a > 0, the parabola opens up

    4. If a < 0, the parabola opens down
    If a = 0, the graph is not a parabola; it is a straight line
  4. If |a| is big, the graph is narrow

    5. If |a| is small, the graph is wide
  5. The y-intercept is (0,c)
  6. The x-intercepts are the zeros of the function.
    1. If the discriminant is greater than 0, then there are 2 x-intercepts
    2. If the discriminant is equal to 0, then there is 1 x-intercept
    3. If the discriminant is less than 0, then there are no x-intercepts
To graph y = a(x - h)2 + k, use the following information:

The vertex of the parabola is (h,k)

Plot some points, using the fact that the parabola is symmetric to the line x = h

To graph y = (x + a)(x + b), use the following information:

Find the x-intercepts by letting y be 0 and solving for x

Find the y-intercepts by letting x be 0 and solving for y

The x-value of the vertex is the average of the two zeros. Substitute that x-value into the equation to find the y-value that goes with it.

Problems

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