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How to Graph a Negative Parabola

A parabola is similar in shape to an elongated circle, an ellipse, with one open end. This characteristic U shape makes a parabola particularly easy to identify, with variations only in the steepness of the graph, the direction of the opening of the graph and its vertical and horizontal translations. You typically define a parabola by a "standard form" equation ax^2 + bx + c, where a, b and c are constant coefficients. You can also express a parabola in "vertex form," a(x - h)^2 + k, where a is a constant coefficient and (h, k) is the vertex point of the parabola. A negative parabola is one that opens toward negative infinity.

Instructions

Standard Form
- 1
  Determine the vertex point of the parabola in standard form: y = ax^2 + bx + c by substituting the numeric values of "a" and "b" into the expression, x = -b / 2a. For example, the x- coordinate of the vertex of the standard form equation -x^2 + 6x + 8, where a = -1 and b = 6 is: x = -(6) / 2(-1) = -6 / -2 = 3. Substitute the value into the equation to find the y-coordinate. For example, y = -(3)^2 + 6(3) + 8 = -9 + 18 + 8 = 17. So the vertex is (3, 17).
- 2
  Plot the vertex onto a coordinate plane.
- 3
  Substitute several x-values into the equation on both sides of the vertex point to get a general idea of the shape of the parabola. For example, for the parabola defined by the standard form equation y = -x^2 + 6x + 8, with vertex (3, 17), substitute x-values such as x = --5, x = -1, x = 0, x = 2, x = 4, x = 8 and x = 10. Solving the equation for x = -5 finds: y(-5) = -(-5)^2 + 6(-5) + 8 = -25 - 30 + 8 = -47. This equates to the coordinate point (-5, -47). Similarly, the points at the remaining x-values are: y(-1) = 1, y(0) = 8, y(2) = 24, y(4) = 16, y(8) = -8, y(10) = -32.
- 4
  Plot all of the points you just found onto the graph.
- 5
  Connect the points together with a smooth curve, moving to the right from the leftmost point. The result should resemble an upside-down U.
Vertex Form
- 6
  Examine the equation of the parabola in vertex form: y = a(x - h)^2 + k where the vertex is (h, k). The value of "h" will be the opposite of what it is in the equation. For example, the parabolic equation y = -3(x + 2)^2 + 5 has a vertex at the point (-2, 5).
- 7
  Plot the vertex point onto a coordinate plane.
- 8
  Substitute several x-values into the equation on both sides of the vertex point to get a general idea of the shape of the parabola. For example, for the parabola defined by the vertex form equation y = -3(x + 2)^2 + 5, with vertex (-2, 5), substitute x-values such as x = -10, x = -5, x = -3, x = -1, x = 0, x = 5 and x = 10. Solving the equation for x = -10 finds: y(-10) = -3(-10 + 2)^2 + 5 = -3(64) + 5 = -192 + 5 = -187. This equates to the coordinate point (-10, -187). Similarly, the points at the remaining x-values are: y(-5) = -22, y(-3) = 2, y(-1) = 2, y(0) = -7, y(5) = -142, y(10) = -427.
- 9
  Plot all of the points you just found onto the graph.
- 10
  Connect the points together with a smooth curve, moving to the right from the leftmost point. The result should resemble an upside-down U.

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