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Quadratic Expressions in Different Forms

7.17.1 Quadratic Expressions in Different Forms

7.17.1 • Quadratic Expressions in Different Forms

Warm Up

Graphs of two parabolas on coordinate plane with unlabeled axes.

Here are graphs representing the functions f f and g g given by f ( x ) = x ( x + 6 ) f ( x ) = x ( x + 6 ) and g ( x ) = x ( x + 6 ) + 4 g ( x ) = x ( x + 6 ) + 4 .

1.

Which graph represents f ( x ) = x ( x + 6 ) f ( x ) = x ( x + 6 ) ? How do you know?

Compare your answer:

The lower graph (which goes through the origin) is the graph of f f .

  • The value of g g is always 4 greater than the value of f f , no matter which value of x x we choose.
  • Rewriting the equation in standard form gives f ( x ) = x 2 + 6 x f ( x ) = x 2 + 6 x . This form shows that the y y -intercept of the graph representing f f is ( 0 , 0 ) ( 0 , 0 ) .
  • The expression that defines function f f is in factored form. It shows that the two x x -intercepts are ( 0 , 0 ) ( 0 , 0 ) and ( − 6 , 0 ) ( − 6 , 0 ) .
2.

Which graph represents g ( x ) = x ( x + 6 ) + 4 g ( x ) = x ( x + 6 ) + 4 ? How do you know?

Compare your answer:

Answer: The lower graph (which goes through the origin) is the graph of f f and the upper one is the graph of g g .

  • The value of g g is always 4 greater than the value of f f , no matter which value of x x we choose.
  • Rewriting the equation in standard form gives g ( x ) = x 2 + 6 x + 4 g ( x ) = x 2 + 6 x + 4 .This form shows that the y y -intercept of the graph representing g g is ( 0 , 4 ) ( 0 , 4 ) .
3.

Where does the graph of f f meet the x x -axis? Explain how you know.

Compare your answer:

x = − 6 x = − 6 and x = 0 x = 0 , because these are the two values of x x that make x ( x + 6 ) x ( x + 6 )   equal to 0.