color(blue)("Determine the structure of the vertex form")Determine the structure of the vertex form
Multiply out the brackets giving:
y=2x^2+10x+13x+65y=2x2+10x+13x+65
y=2x^2+23x+65" "y=2x2+23x+65 ...................................(1)
write as:
y=2(x^2+23/2x)+65y=2(x2+232x)+65
What we are about to do will introduce an error for the constant. We get round this by introducing a correction.
Let the correction be k then we have
color(brown)(y=2(x+23/4)^2+k+65" ")y=2(x+234)2+k+65 ..................................(2)
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To get to this point I moved the square from x^2x2 to outside the brackets. I also multiplied the coefficient of 23/2x232x by 1/212 giving the 23/4234 inside the brackets.
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color(blue)("Determine the value of the correction")Determine the value of the correction
We need the values of a point for substitution so that k can be calculated.
Using equation (1) set x=0x=0 giving
y=2(0)^2+23(0)+65 => y=65y=2(0)2+23(0)+65⇒y=65
So we have our ordered pair of (x,y)->(0,65)(x,y)→(0,65)
Substitute this into equation (2) giving:
cancel(65)=2(0+23/4)^2+k+cancel(65)" ".................................(2_a)
k=-529/8
y=2(x+23/4)^2-529/8+65" "..................................(3)
But" "65-529/8 = 9/8
Substitute into equation (3) gives:
color(blue)("vertex form "->" "y=2(x+23/4)^2+9/8)
color(brown)("Note that "(-1)xx23/4 = -5 3/4 ->" axis if symmetry")
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