How do you write #f(x) = -4x^2 - 16x + 3# in vertex form?

Answer 1

#f(x)=-4(x+2)^2+19#

#"for the standard form of a parabola " y=ax^2+bx+c#
#"the x-coordinate of the vertex is " x_(color(red)"vertex")=-b/(2a)#
#y=-4x^2-16x+3" is in standard form"#
#"with " a=-4,b=-16,c=3#
#rArrx_(color(red)"vertex")=-(-16)/(-8)=-2#
#"substitute into f(x) for y-coordinate"#
#rArry_(color(red)"vertex")=-4(-2)^2-16(-2)+3=19#
#rArrcolor(magenta)"vertex "= (-2,19)#
#"the equation of a parabola in "color(blue)"vertex form"# is.
#• y=a(x-h)^2+k#

where ( h , k ) are the coordinates of the vertex and a is a constant.

#"here " (h,k)=(-2,19)" and "a=-4#
#rArry=-4(x+2)^2+19larrcolor(red)" in vertex form"#
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Answer 2

To write the function ( f(x) = -4x^2 - 16x + 3 ) in vertex form, follow these steps:

  1. Complete the square for the quadratic term ( -4x^2 - 16x ) by factoring out the coefficient of ( x^2 ) and halving the coefficient of ( x ) before squaring it.

  2. Add and subtract the square of half the coefficient of ( x ) inside the parentheses to maintain the equivalence of the expression.

  3. Rewrite the expression to isolate the squared term and simplify.

  4. The resulting expression will be in the form ( f(x) = a(x - h)^2 + k ), where ( (h, k) ) represents the vertex of the parabola.

Applying these steps to the function ( f(x) = -4x^2 - 16x + 3 ):

  1. Complete the square for the quadratic term ( -4x^2 - 16x ): [ -4x^2 - 16x = -4(x^2 + 4x) ]

  2. Halve the coefficient of ( x ) and square it: [ (-4x^2 - 16x) = -4(x^2 + 4x + 4 - 4) ]

  3. Rewrite the expression to isolate the squared term and simplify: [ -4(x^2 + 4x + 4 - 4) = -4((x + 2)^2 - 4) ]

  4. Expand and simplify: [ -4((x + 2)^2 - 4) = -4(x + 2)^2 + 16 ]

Thus, the function ( f(x) = -4x^2 - 16x + 3 ) in vertex form is ( f(x) = -4(x + 2)^2 + 16 ).

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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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