5.1.3 Standard Form, Special Products, Vertex

For the polynomial y = ax 2 + bx + c, we have a special name, known as a Quadratic Quadratics will be of the form as a parabola

Standard Form For a quadratics, and any polynomials in general, we will write them in the future in standard form = descending order of powers Example. y = 4x 2 + 8x + 12 All polynomials should be written in this form in the future!

Example. Write the following function in standard form. y = (x + 3)(x – 7)

Example. Write the following function in standard form. y = -(x - 9)(x + 2)

Example. Write the following function in standard form. y = -3(x + 1)(x - 5)

Example. Write the following function in standard form. z = -(5y + 1)(2y - 2) + 9 PEMDAS!

Vertex For any given quadratic, their shape of a parabola will have what is known as the vertex Vertex = highest or lowest point of a quadratic – Includes both an x and y value – Ordered pair point

To find the vertex, we can use a simple formula to help us For the quadratic y = ax 2 + bx + c; X-coordinate = -b/2a (opposite b over 2 a) Y – coordinate = f(-b/2a); plug the x-value back in

Example. Find the vertex for the quadratic y = x 2 – 6x + 5 What is the b-value? What is the a-value?

Example. Find the vertex for the quadratic y = x 2 - 8x + 15 What is the b-value? What is the a-value?

Special Product As we continue with polynomials, we will look at many cases of “special products” One particular case is the perfect square binomial (ax + b) 2 = a 2 x 2 + 2abx + b 2 (ax - b) 2 = a 2 x 2 - 2abx + b 2 “Square the first, twice the product, square the last”

Example. Write the following function in standard form. y = (x – 1) 2 - 3

Example. Write the following function in standard form. y = -(x + 2) 2

Find the following for each problem; standard form and the vertex 1) (x + 10)(x – 4) 2) (x – 4) 2 3) (2x – 6)(x + 8) 4) (x – 9) 2

S Assignment Pg odd (find the vertex ONLY) all