How do you find the vertex and the intercepts for #y= (x-4) (x+2)#?

Answer 1

See below

For the vertex, we need to get our equation in standard form. Using FOIL, it can be rewritten as #y=x^2-2x-8#. To find its x-coordinate, use the formula #x=-b/(2a)#:
#x=(-(-2))/(2(1))=2/2=1#

The y-coordinate is found by plugging our answer back into the equation:

#y=(1)^2-2(1)-8# #y=1-2-8=-7#
Vertex: #(1,-7)#
By "intercepts", I believe you mean both the x-intercepts and the y-intercept. For the x-intercepts, we can use the factored equation, set #y# to #0#, and solve:
#0=(x-4)(x+2)# #x-4=0# #x=4# #x+2=0# #x=-2#
X-intercepts: #(4,0), (-2,0)#
To find the y-intercept, we will use the standard form of our equation, set #x# to #0#, and solve:
#y=0^2-2(0)-8# #y=-8#
Y-intercept: #(0,-8)#

Hope this helped!

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Answer 2

To find the vertex and intercepts for ( y = (x - 4)(x + 2) ):

  1. Vertex: The vertex of a quadratic function in the form ( y = ax^2 + bx + c ) is at ( x = -\frac{b}{2a} ). In this case, ( a = 1 ) and ( b = -2 ), so the x-coordinate of the vertex is ( x = -\frac{-2}{2(1)} = 1 ). Substitute ( x = 1 ) into the equation to find the y-coordinate of the vertex: ( y = (1 - 4)(1 + 2) = (-3)(3) = -9 ). Therefore, the vertex is at (1, -9).

  2. x-intercepts (or roots): Set ( y = 0 ) and solve for ( x ): ( (x - 4)(x + 2) = 0 ). This equation is satisfied when either ( x - 4 = 0 ) or ( x + 2 = 0 ). Solving these gives ( x = 4 ) and ( x = -2 ). So, the x-intercepts are at ( x = 4 ) and ( x = -2 ).

  3. y-intercept: Substitute ( x = 0 ) into the equation: ( y = (0 - 4)(0 + 2) = (-4)(2) = -8 ). Therefore, the y-intercept is at (0, -8).

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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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