What is the vertex of #y=3(x-3)^2-x^2+12x - 15#?

Answer 1

#"Vertex "-> (x,y)->(3/2,15/2)#

#color(blue)("Method:")#
First simplify the equation so that it is in standard form of:
#color(white)("xxxxxxxxxxx)y=ax^2+bx+c#

Change this into the form:
#color(white)("xxxxxxxxxxx)y=a(x^2+b/ax)+c# This is NOT vertex form

Apply # -1/2xxb/a= x_("vertex")#

Substitute #x_("vertex")# back into the standard form to determine
#y_("vertex")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given:#color(white)(.....) y=3(x-3)^2-x^2+12x-15#

#color(blue)("Step 1")#

#y=3(x^2-6x+9)-x^2+12x-15#

#y=3x^2-18x+27-x^2+12x-15#

#y=2x^2-6x+12# ...........................................(1)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#

Write as: #y=2(x^2-3x)+12#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#

#color(green)(x_("vertex") = (-1/2)xx(-3)=+3/2)#.........................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4")#

Substitute value at (2) into equation (1) giving:

#y_("vertex")=2(3/2)^2-6(3/2)+12#

#y_("vertex")=18/4-18/2+12#

#y_("vertex")=18/4-36/4+12#

#color(green)(y_("vertex")=-9/2+12=15/2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#"Vertex "-> (x,y)->(3/2,15/2)->(1 1/2, 7 1/2)#

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Answer 2

The vertex of the function (y = 3(x - 3)^2 - x^2 + 12x - 15) is (-3, 6).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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