INTERCEPT FORM EQUATION OF A PARABOLA

Examples 1-4 : Write the following equations of parabolas in intercept form.

Example 1 :

Write the following equation of parabola in intercept form.

y = x² - 5x + 6

Solution :

y = x² - 5x + 6

Factor the quadratic expression x² - 5x + 6.

In the quadratic expression above, the coefficient of x² is 1. So, get two factors of the constant term '+6' such that the sum of the two factors is equal to the coefficient of x, that is '-5'.

Then the two factors of '+6' are '-2' and '-3'.

Now, split the middle term -5x using the two factors -2 and -3.

x² - 5x + 6 :

= x² - 2x - 3x + 6

= x(x - 2) - 3(x - 3)

= (x - 2)(x - 3)

Therefore, the intercept form equation of the parabola is

y = (x - 2)(x - 3)

Example 2 :

y = x² - 5x - 24

Solution :

y = x² - 5x - 24

Factor the quadratic expression x² - 5x - 24.

In the quadratic expression above, the coefficient of x² is 1. So, get two factors of the constant term '-24' such that the sum of the two factors is equal to the coefficient of x, that is '-5'.

Then the two factors of '-24' are '-8' and '+3'.

Now, split the middle term -5x using the two factors -8 and +3.

x² - 5x - 24 :

= x² - 8x + 3x - 24

= x(x - 8) + 3(x - 8)

= (x - 8)(x + 3)

Therefore, the intercept form equation of the parabola is

y = (x - 8)(x + 3)

Example 3 :

y = 2x² - 12x + 16

Solution :

y = 2x² - 12x + 16

Factor 2.

y = 2(x² - 6x + 8)

Factor the quadratic expression x² - 6x + 8.

In the quadratic expression above, the coefficient of x² is 1. So, get two factors of the constant term '+8' such that the sum of the two factors is equal to the coefficient of x, that is '-6'.

Then the two factors of '+8' are '-2' and '-4'.

Now, split the middle term -6x using the two factors -2 and -4.

x² - 6x + 8 :

= x² - 2x - 4x + 8

= x(x - 2) - 4(x - 2)

= (x - 2)(x - 4)

Therefore, the intercept form equation of the parabola is

y = 2(x - 2)(x - 3)

Example 4 :

y = 2x² + 11x + 12

Solution :

y = 2x² + 11x + 12

Here, the coefficient of x² is 2. But, 2 can not be factored from the expression 2x² + 11x + 12 as we have done in the previous example.

So, we have to factor the expression 2x² + 11x + 12 as it is.

Multiply the coefficient of x² and the constant term 12.

2x12 = 24

Now, get two factors of '+24' such that the sum of the two factors is equal to the coefficient of x, that is '+11'.

Then the two factors of '+24' are '+3' and '+8'.

Now, split the middle term +11x using the two factors +3 and +8.

2x² + 11x + 12 :

= 2x² + 8x + 3x + 12

= 2x(x + 4) + 3(x + 4)

= (x + 4)(2x + 3)

= (2x + 3)(x + 4)

= 2(x + 3/2)(x + 4)

Therefore, the intercept form equation of the parabola is

y = = 2(x + 3/2)(x + 4)

Example 5 :

Find the vertex of the parabola :

y = -3(x + 2)(x - 6)

Solution :

y = -3(x + 2)(x - 6)

Substitute y = 0 to find the x-intercepts of the parabola above, 

-3(x + 2)(x - 6) = 0

Divide both sides by -3.

-3(x + 2)(x - 6) = 0

x + 2 = 0  or  x - 6 = 0

x = -2  or  x = 6

x-coordinate of the vertex :

x = (-2 + 6)/2

x = 4/2

x = 2

y-coordinate of the vertex :

Substitute x = 2 in y = -3(x + 2)(x - 6).

y = -3(2 + 2)(2 - 6)

y = -3(4)(-4)

y = 48

Vertex of the parabola is (2, 48).

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