SOLUTION: If you know the vertex of the graph of a quadratic function, can you specify the range of the function? If so, what is the range? If not what additional information do you need?

Question 20923: If you know the vertex of the graph of a quadratic function, can you specify the range of the function? If so, what is the range? If not what additional information do you need?
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
If you know the vertex of the graph of a quadratic function, can you specify the range of the function? If so, what is the range? If not what additional information do you need?
LET US SEE THIS FIRST VISUALLY BY PLOTTING SOME GRAPHS FOR
1....Y=(X-2)^2+4
2....Y=4-(X-2)^2
3....Y=(X-2)^2-4

SO YOU FIND THAT WHEN QUADRATIC EQUATION GIVES US A VERTEX ,AS THE NAME IMPLIES ,WE ARE GETTING A PEAK VALUE OR TROUGH VALUE FOR THE FUNCTION Y AT THAT POINT .DEPENDING ON WHETHER IT IS PEAK (MAXIMUM)OR TROUGH (MINIMUM) ,ONE BOUNDARY FOR Y IS FIXED ,THE OTHER BOUNDARY BEING PLUS INFINITY OR MINUS INFINITY AGAIN DEPENDING ON EHETHER WE GOT A PEAK (MAXIMUM)OR TROUGH (MINIMUM).
SO IN THIS WAY THE RANGE OF Y GETS FIXED.HENCE WE NEED THE VERTEX ,MORE PRECISELY THE Y COORDINATE OF THE VERTEX AND ITS NATURE..... PEAK (MAXIMUM)OR TROUGH (MINIMUM)...TO DEFINE THE RANGE OF THE FUNCTION.
YOU CAN SEE IT ALGEBRAICALLY ALSO AS A PERFECT SQUARE IS ALWAYS POSITIVE OR ZERO....ITS MINIMUM VALUE CAN BE ZERO.....HENCE IT WILL CONTRIBUTE TO MAXIMIXE OR MINIMISE THE FUNCTION (VALUE OF Y)AT THAT POINT DEPENDING ON THE SIGN INFRONT OF THE PERFECT SQUARE BEING NEGATIVE OR POSITIVE RESPECTIVELY.IN EXAMPLES 1 AND 3 IT IS POSITIVE +(X-2)^2. HENCE VERTEX OR Y VALUE BECOMES MINIMUM THERE SINCE YOU GET THE LOWEST VALUE BY ADDING THE LEAST VALUE OF ZERO. IN EXAMPLE 2 IT IS NEGATIVE -(X-2)^2.HENCE VERTEX OR Y VALUE IS MAXIMUM THERE SINCE YOU GET THE HIGHEST VALUE BY SUBTRACTING THE LEAST VALUE OF ZERO.
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