`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.
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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.
| Quadratic Equation in intercept form | X-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`
thank you so much!
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ReplyDeletewait... what does "a" stand for in the equation?
ReplyDeleteIf a is > 0, the graph opens upward. If a < 0, the graph opens downward.
Delete"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.
DeleteCan every quadratic be written in this form. And what if the parabola doesn't touch the x-intercept?
ReplyDeleteAll 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.
DeleteHope that helps
In general, any quadratic function can be written in the intercept form.
ReplyDeletey = 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)
How do you find the y-intercept?
ReplyDeleteTo find the y-intercept, plug in x = 0 in the quadratic equation. For example,
Delete`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)`