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Three Forms of Quadratic Expressions

7.16.1 Three Forms of Quadratic Expressions

7.16.1 • Three Forms of Quadratic Expressions

Warm Up

Expressions in different forms can be used to define the same function. Here are three ways to define a function f f .

Standard form Factored form Vertex form
f ( x ) = x 2 − 4 x + 3 f ( x ) = x 2 − 4 x + 3 f ( x ) = ( x − 3 ) ( x − 1 ) f ( x ) = ( x − 3 ) ( x − 1 ) f ( x ) = ( x − 2 ) 2 − 1 f ( x ) = ( x − 2 ) 2 − 1
f ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c f ( x ) = a ( x − r 1 ) ( x − r 2 ) f ( x ) = a ( x − r 1 ) ( x − r 2 ) f ( x ) = a ( x − h ) 2 + k f ( x ) = a ( x − h ) 2 + k

Which form would you use if you want to find the following features of the graph of f f ? Be prepared to explain your reasoning.

1.

The x x -intercepts

Factored form. In factored form, we can see that the x x -intercepts, or roots, are located at ( r 1 , 0 ) ( r 1 , 0 ) and ( r 2 , 0 ) ( r 2 , 0 ) .

2.

The vertex

Vertex form. In vertex form, we can see that the graph has a vertex located at the point defined by ( h , k ) ( h , k ) .

3.

The y y -intercept

In standard form, we can see that the graph intersects the y y -axis at 3, but in any form, we can pretty easily find the y y -intercept when the input is 0.