Question

Write y=x⁽²⁾-2x+20 in vertex form. Then identify the vertex.

Answer 1

The quadratic equation y = x² - 2x + 20 is rewritten in vertex form y = (x - 1)² + 19 by completing the square, and the vertex of the parabola is identified as (1, 19).

Step-by-step explanation:

To rewrite the quadratic equation y = x² - 2x + 20 in vertex form, we need to complete the square. Vertex form of a quadratic equation is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

First, factor out the coefficient of x² (which is 1 in this case, so this step is not necessary here). The equation remains y = x² - 2x + 20. Now, we need to create a perfect square trinomial from the x-terms. The perfect square trinomial is obtained by taking half of the coefficient of x, squaring it, and adding it to and subtracting it from the equation.

Half of the coefficient of x is -1 (since -2 divided by 2 equals -1), and squaring it gives us 1. Add and subtract this number within the equation:

y = (x² - 2x + 1) + 20 - 1

y = (x - 1)² + 19

The equation is now in vertex form. The vertex of the parabola is (1, 19).

Answer 2

The vertex form of the parabola y = x² - 2x + 20 is y = ( x - 1 )² + 19 and the vertex is ( 1, 19).

The vertex form of a quadratic function is expressed as:

y = a( x - h )² + k

Where (h, k) is the vertex of the parabola

Given the quadratic equation in the question:

y = x² - 2x + 20

To write the quadratic equation y = x² - 2x + 20 in vertex form, first, we complete the square:

y = x² - 2x + 20

y = ( x² - 2x ) + 20

Add and subtract 1 in the parenthesis:

y = ( x² - 2x + 1 - 1 ) + 20

Factoring, we get:

y = ( x - 1 )² - 1 + 20

y = ( x - 1 )² + 19

Hence, the vertex form of the equation is y = ( x - 1 )² + 19.

Now compare to y = a( x - h )² + k:

Note that; ( h, k ) is the vertex.

y = ( x - 1 )² + 19

Vertex ( h, k ) = ( 1, 19)

Therefore, the vertex of the parabola is ( 1, 19).