Convert Quadratic Functions From Standard Form To Intercept Form Or Vertex Form Worksheet By Nghi H Nguyen With Answers printable pdf download

Convert Quadratic Functions From Standard Form To Intercept Form Or Vertex Form Worksheet By Nghi H Nguyen With Answers

Convert quadratic functions from Standard Form

to Intercept Form or Vertex Form

(By Nghi H Nguyen – Dec 03, 2014)

The Quadratic Function in Intercept Form

The graph of the quadratic function in standard form f(x) = ax

+ bx + c is a parabola that may

intercept the x-axis at 2 points, unique point, or no point at all. This means a quadratic equation

f(x) = ax

+ bx + c = 0 may have 2 real roots, one double real root, or no real roots at all

(complex roots).

We can write f(x) in the form: y = a(x² + b/a x + c/a) (1).

Recall the development of the quadratic formula:

x² + bx/a + (b²/4a² - b²/4a²) + c/a = 0

(x + b/2a)² - (b² - 4ac)/4a² = 0

(x + b/2a)² - d²/4a² = 0.

(Call d² = b² - 4ac)

(x + b/2a + d/2a)(x + b/2a – d/2a) = 0 (2)

Replace this expression (2) into the equation (1), we get the quadratic function f(x) written in

intercept form:

f(x) = a(x – x1)(x – x2)

(x1 and x2 are the 2 x-intercepts, or roots of f(x) = 0)

f(x) = a(x + b/2a + d/2a)(x + b/2a – d/2a) (3)

The Quadratic Formula in Intercept Form

. From the equation (3), we deduct the

formula:

x = -b/2a ± d/2a (4)

In this formula, x being the 2 roots of the quadratic equation f(x) = 0:

The quantity (–b/2a) represents the x-coordinate of the parabola axis.

The 2 quantities (d/2a) and (-d/2a) represent the 2 distances AB and AC from the

parabola axis to the two x-intercepts (real roots) of the parabola.

The quantity d can be zero, a number (real or radical), or imaginary number.

If d = 0: there is a double root at x = -b/2a.

If d is a number (real or radical), there are two x-intercepts, meaning two real roots.

If d is imaginary, there are no real roots. The parabola doesn’t intercept the x-axis.

Convert Quadratic Functions From Standard Form To Intercept Form Or Vertex Form Worksheet By Nghi H Nguyen With Answers