How do you find the intercept and vertex of #y= -3(x-1)²-1#?

Answer 1

Vertex#->(x,y)=(1,-1)#
#y_("intercept")=-4#

No #x_("intercept") in RR#

#color(blue)("Determine the vertex")#

This is the Vertex Form of a quadratic equation so you can virtually directly read off the coordinates of the vertex.

#y=-3(xcolor(red)(-1))^2color(blue)(-1)#
#x_("vertex")=(-1)xx color(red)((-1)) = +1# #y_("vertex")=color(blue)(-1)#
Vertex#->(x,y)=(1,-1)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine the y intercept")#
Set #x=0# giving
#y_("intercept")=-3(0-1)^2-1" "=" "-4# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine the x intercept")#
Set #y=0# giving
#0=-3(x-1)^2-1" " larr # add 1 to both sides
#=>1=-3(x-1)^2" "larr #multiply both sides by (-1)
#=>-1=+3(x-1)^2" "larr #divide both sides by 3
#=>-1/3=(x-1)^2" "larr # square root both sides
#=>+-sqrt(-1/3)=x-1" "larr # add 1 to both sides
#x=1+-sqrt(-1/3)" " rarr x in CC#

As the determinant is negative the curve does not cross the x-axis nor is the axis a tangent to the curve.

Thus the roots are in the number range of 'Complex Numbers' '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)(" The question lists intercepts as 'singular' thus the x-axis roots are not required")#
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Answer 2

To find the intercept and vertex of ( y = -3(x-1)^2 - 1 ):

  1. Intercept: Substitute ( x = 0 ) into the equation and solve for ( y ).

    ( y = -3(0-1)^2 - 1 )

  2. Vertex: Use the formula ( x = \frac{-b}{2a} ) to find the x-coordinate of the vertex. Then substitute this x-value into the equation to find the y-coordinate.

    ( x = \frac{-(-1)}{2(-3)} )

    ( y = -3\left(\frac{-1}{2}\right)^2 - 1 )

    ( y = -\frac{5}{2} )

So, the intercept is at (0, -2) and the vertex is at (1, -2.5).

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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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