How can I determine the x and y intercepts of a parabola?

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To find the x and y intercepts of a parabola, you need to understand the standard equations of parabolas and how they relate to the coordinate system.

Finding the Y-Intercept

The y-intercept of a parabola is the point where it intersects the y-axis. To find it, follow these steps:

  1. Set x = 0 in the equation of the parabola. The standard form of a parabola can be given as:
    • y = ax2 + bx + c
  2. Substituting x = 0 into the equation results in:
    • y = a(0)2 + b(0) + c = c
  3. Thus, the y-intercept is the point (0, c).

Finding the X-Intercepts

The x-intercepts of a parabola are the points where it intersects the x-axis. To find them, follow these steps:

  1. Set y = 0 in the equation of the parabola:
    • 0 = ax2 + bx + c
  2. This is a quadratic equation in the standard form. To solve for x, you can use the quadratic formula:
    • x = (-b ± √(b2 – 4ac)) / 2a
  3. Calculate the discriminant (b2 – 4ac):
    • If the discriminant is positive, there are two x-intercepts.
    • If it is zero, there is one x-intercept (the vertex of the parabola).
    • If it is negative, there are no real x-intercepts.

Example

Consider the parabola given by the equation:

y = 2x2 + 3x – 5

Y-Intercept:

Setting x = 0:

y = 2(0)2 + 3(0) – 5 = -5

The y-intercept is (0, -5).

X-Intercepts:

Setting y = 0:

0 = 2x2 + 3x – 5

Using the quadratic formula:

x = (-3 ± √(32 – 4(2)(-5))) / (2(2))

x = (-3 ± √(9 + 40)) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4

  • x = 1
  • x = -2.5

The x-intercepts are (1, 0) and (-2.5, 0).

Now you have the y-intercept and the x-intercepts:

  • Y-Intercept: (0, -5)
  • X-Intercepts: (1, 0) and (-2.5, 0)

Using these methods, you can find the intercepts of any parabola!

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