Intercept form of a quadratic equation:

The intercept form of a quadratic equation is as follows:

`y = a(x - p)(x - q)`

The intercept form of a quadratic equation is the form of a quadratic equation by which you can easily tell the x intercepts of the quadratic equation.

It is thus very useful to convert a quadratic equation to its intercept form in order to get the value of its two x-intercepts.

Meaning of the various symbols in the intercept form of a quadratic equation:

  • `p` and `q` are the x-intercepts (also zeros, solutions, roots)
    • The axis of symmetry can be found out from the values of `p` and `q` by the equation  `x = (p + q)/2`. This is because the axis of symmetry passes through the vertex and hence is in the exact middle of the two x-intercepts. If there is only one x-intercept, or if both x-intercepts are same, then the axis of symmetry is equal to `x = p = q`.
  • `a` is the vertical stretching factor
    • `a` can't be equal to zero, because then the equation will not be a quadratic one
    • If `a` is negative, the parabola faces downwards, and if `a` is positive then the parabola faces upwards
    • If the absolute value of `a`, that is `|a|` is greater than 1 then the parabola gets narrower because it gets stretched vertically and if it is lesser than 1 then the parabola gets wider because it gets stretched vertically by a fractional factor whose denominator is greater than the numerator.
The x-intercepts of parabolas can be obtained from their quadratic equations in intercept forms as follows:

Quadratic Equation in intercept formX-intercepts of its parabola
y = (x - 3)(x - 4)3 and 4
y = x(x - 5)0 and 5
y = (x + 3)(x + 5)-3 and -5

It is also very easy to get the equation of the axis of symmetry of a parabola from the intercept form of a quadratic equation. The axis of symmetry can be calculated with the help of the following formula:
Axis of symmetry, `x = (p + q)/2`
In the above formula, p and q get their values from the intercept form of the quadratic equation.

For example, for the quadratic equation `y = (x - 3)(x - 4)`, the axis of symmetry is :
`x = (3 + 4)/2 = 7/2`

13 comments:

  1. thank you so much!

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    Replies
    1. diz helpud zooooooo MUCh!!...!!!! lolzzz3

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    2. Engrish please

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  2. wait... what does "a" stand for in the equation?

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    1. If a is > 0, the graph opens upward. If a < 0, the graph opens downward.

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    2. "a" is the amtitude. it its 2 the parabola is narrow and if its a fraction then its wider as well. The response above is also true.

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  3. Can every quadratic be written in this form. And what if the parabola doesn't touch the x-intercept?

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    1. All quadratics can't be written in intercept form, only the ones which can be factored, which are not necessarily the ones whose parabolas donot have x-intercepts. Some quadratic equations, for example x^2 + 3x + 1 do have x intercepts but it can't be factored into the intercept form.

      Hope that helps

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  4. In general, any quadratic function can be written in the intercept form.
    y = ax^2 + bx + c
    y = a*(x - x1)(x - x2)
    y = a*(x - b/2a + d/2a)(x - b/2a - d/2a) (1) (call d^2 = b^2 - 4ac)
    This expression (1) is the general intercept form of a quadratic function.
    When the values of a, b, and c are specifically given, then we can find the true intercept form.
    Example: Write y = x^2 + 8x + 15 in intercept form. We have:
    d^2 = 64 - 60 = 4 --> d = 2 and d = -2
    y = (x + 8/2 - 2/2)(x + 8/2 + 2/2)
    y = (x + 3)(x + 5)

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  5. How do you find the y-intercept?

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    1. To find the y-intercept, plug in x = 0 in the quadratic equation. For example,
      `y = 2x^2 + 3x + 1`
      Plugging in x = 0 you get
      `y = 2(0)^2 + 3(0) + 1 = 1`
      Thus its y-intercept is 1

      If a quadratic equation is of the form x = f(y) (that is, it is a function of y) then in order to find the y-intercept, follow the method discussed above in the post with 'y' in place of 'x' and 'x' in place of 'y'. For example, `x = 2y^2 + 3y + 1` can be factored into `x = (2y + 1)(y + 1)`

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