Finding Maximum and Minimum Values in Quadratic Equations

Introduction

• The focus is on determining the maximum and minimum values of a quadratic equation.
• Quadratics are expressed in the form ( ax^2 + bx + c ).

Identifying Coefficients

• Important to identify coefficients:
  • ( a = -2 )
  • ( b = 8 )
  • ( c = 7 )

Test for Max/Min Point

• Determine if the graph has a max or min point based on the value of ( a ):
  • If ( a < 0 ), the graph has a maximum point (opens down).
  • If ( a > 0 ), the graph has a minimum point (opens up).
• For this problem, ( a = -2 ), hence the graph has a maximum point.

Finding the Vertex

• The vertex is the key "point" of the quadratic, where it achieves its maximum or minimum value.
• Formula for finding the vertex:
  1. ( x ) coordinate: ( x = -\frac{b}{2a} )
  2. ( x = -\frac{8}{2 \times -2} = 2 )
• To find the ( y ) coordinate, substitute ( x = 2 ) back into the equation:
  • ( y = (-2)(2^2) + 8 \cdot 2 + 7 )
  • ( y = -8 + 16 + 7 )
  • ( y = 15 )

Conclusion

• The maximum value of the quadratic is ( y = 15 ) when ( x = 2 ).
• The vertex is at ( (2, 15) ).
• The maximum point is significant as it represents the highest value the graph will achieve.

Review

• Always start by identifying ( a, b, ) and ( c ).
• Use the test ( a < 0 ) or ( a > 0 ) to determine max/min.
• Calculate the vertex to find the maximum/minimum value of the quadratic.
• Remember the y-value at the vertex gives the max/min value of the function.