You can find information about the quadratic function f ( x ) = a x 2 + b x + c f open x close equals eh , x squared , plus b x plus c  from the coefficients a and b, and from the constant term c.

Take Note Properties: Quadratic Function in Standard Form

• The graph of f ( x ) = a x 2 + b x + c , a ≠ 0 , f open x close equals eh , x squared , plus b x plus c comma eh not equal to 0 comma  is a parabola.
• If a > 0 , eh greater than 0 comma  the parabola opens upward. If a < 0 , eh less than 0 comma  the parabola opens downward.
• The axis of symmetry is the line x = − b 2 a . x equals negative , fraction b , over 2 eh end fraction , .
• The x-coordinate of the vertex is − b 2 a . negative , fraction b , over 2 eh end fraction , .  The y-coordinate of the vertex is the y-value of the function for x = − b 2 a x equals negative , fraction b , over 2 eh end fraction  or y = f ( − b 2 a ) . y equals f . open . negative , fraction b , over 2 eh end fraction . close . .
• The y-intercept is (0, c).

y = a x 2 + b x + c , a > 0 y equals eh , x squared , plus b x plus c comma eh greater than 0

y = a x 2 + b x + c , a < 0 y equals eh , x squared , plus b x plus c comma eh less than 0

Here's Why It Works You can expand the vertex form of a quadratic function to determine properties of the graph of a quadratic function written in standard form.

f ( x ) = a ( x − h ) 2 + k   = a ( x 2 − 2 h x + h 2 ) + k   = a x 2 − 2 a h x + a h 2 + k   = a x 2 + ( − 2 a h ) x + ( a h 2 + k ) table with 4 rows and 2 columns , row1 column 1 , f open x close , column 2 equals eh . open x minus h close squared . plus k , row2 column 1 , , column 2 equals eh open , x squared , minus 2 h x plus , h squared , close plus k , row3 column 1 , , column 2 equals eh , x squared , minus 2 eh h x plus eh , h squared , plus k , row4 column 1 , , column 2 equals eh , x squared , plus open negative 2 eh h close x plus open eh , h squared , plus k close , end table

Compare to the standard form, f ( x ) = a x 2 + b x + c . f open x close equals eh , x squared , plus b x plus c .

a = a a in standard form is the same as a in vertex form.

b = − 2 a h − b 2 a = h Solve for  h . table with 2 rows and 2 columns , row1 column 1 , b , column 2 equals negative 2 eh h , row2 column 1 , negative , fraction b , over 2 eh end fraction , column 2 equals h , column 3 cap solve for . h . , end table

Since, h = − b 2 a , h equals negative , fraction b , over 2 eh end fraction , comma  the axis of symmetry is x = − b 2 a x equals negative , fraction b , over 2 eh end fraction  and the vertex is ( h , k ) = ( − b 2 a , f ( − b 2 a ) ) . open , h comma k , close . equals . open . negative , fraction b , over 2 eh end fraction , comma f . open . negative , fraction b , over 2 eh end fraction . close . close . .

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