Rewrite each equation in vertex form by completing the square. then identify the vertex.

Rewrite each equation in vertex form by completing the square. then identify the vertex.


[tex]Rewrite each equation in vertex form by completing the square. then identify the vertex.[/tex]

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  1. ANSWER

    Vertex form:

    [tex]y = 2( {x - 4)}^{2} - 27[/tex]

    Vertex:

    V(4,-27)

    EXPLANATION

    The given function is

    [tex]y = 2 {x}^{2} - 16x + 5[/tex]

    Complete the square as follows:

    [tex]y = 2( {x}^{2} - 8x) + 5[/tex]

    [tex]y = 2( {x}^{2} - 8x + 16) + 5 - 2 \times 16[/tex]

    [tex]y = 2( {x - 4)}^{2} + 5 - 32[/tex]

    The vertex form is:

    [tex]y = 2( {x - 4)}^{2} - 27[/tex]

    The vertex is:

    V(4,-27)

  2. The vertex form of the given equation is f(x) = (x+7/2)²+(-61/4)  where vertex is (-7/2,-61/4).

    Step-by-step explanation:

    We have given a quadratic equation in standard form.

    f (x)=  x²+7x-3

    We have to rewrite given equation in vertex form.

    y  = a(x-h)²+k is vertex form of equation where (h,k) is vertex of equation.

    We will use method of completing square to solve this.

    Adding and subtracting  (7/2)²  to above equation, we have

    f(x) =  x²+7x-3+(7/2)²-(7/2)²

    f(x) = x²+7x+(7/2)²-3-(7/2)²

    f(x) = (x+7/2)²-3-49/4

    f(x) = (x+7/2)²+(-12-49)/4

    f(x) = (x+7/2)²+(-61/4)

    Hence, The vertex form of the given equation is f(x) = (x+7/2)²+(-61/4)  where vertex is (-7/2,-61/4).

  3. ANSWER

    Vertex form;

    [tex]y = 3( {x + \frac{3}{2} })^{2} - \frac{35}{4}[/tex]

    Vertex

    [tex]V( - \frac{3}{2} , - \frac{35}{4} )[/tex]

    EXPLANATION

    Given:

    [tex]f(x) = 3 {x}^{2} + 9x - 2[/tex]

    We complete the square as follows:

    [tex]y = 3( {x}^{2} + 3x) - 2[/tex]

    [tex]y = 3( {x}^{2} + 3x + \frac{9}{4} ) - 2 - 3 \times \frac{9}{4}[/tex]

    The vertex form is:

    [tex]y = 3( {x + \frac{3}{2} })^{2} - \frac{35}{4}[/tex]

    The vertex is

    [tex]V( - \frac{3}{2} , - \frac{35}{4} )[/tex]

  4. The vertex form of the given equation is f(x) = 3(x+3/2)²+(19/4)  where vertex is (-3/2,19/4).

    Step-by-step explanation:

    We have given a quadratic equation in standard form.

    f(x)= 3x²+9x-2

    We have to rewrite given equation in vertex form.

    f (x) = a(x-h)²+k is vertex form of equation where (h,k) is vertex of equation.

    We will use method of completing square to solve this.

    Adding and subtracting  (9/2)²  to above equation, we have

    f(x)   = 3(x²+3x)-2

    f(x)   = 3(x²+3x+(3/2)² ) -2+3(3/2)²

    f(x)   = 3(x²+3x+(3/2)² ) -2+3(9/4)

    f(x) =  3(x+3/2)²-2+27/4

    f(x) =  3(x+3/2)²+(-8+27)/4

    f(x) =  3(x+3/2)²+(19/4)

    Hence, The vertex form of the given equation is f(x) = 3(x+3/2)²+(19/4)  where vertex is (-3/2,19/4).

  5. The vertex form of the given equation is y  = (x+4)²-11 where vertex is (-4,-11).

    Step-by-step explanation:

    We have given a quadratic equation in standard form.

    y = x²+8x+5

    We have to rewrite given equation in vertex form.

    y  = (x-h)²+k is vertex form of equation where (h,k) is vertex of equation.

    We will use method of completing square to solve this.

    Adding and subtracting  (4)²  to above equation, we have

    y = x²+8x+5 +(4)²-(4)²

    y = x²+8x+(4)²+5-(4)²

    y = (x)²+2(x)(4)+(4)²+5-16

    y  = (x+4)²-11

    Hence, the vertex form of the given equation is y  = (x+4)²-11 where vertex is (-4,-11).

  6. ANSWER

    Vertex form:

    [tex]y = ( {x + 4)}^{2} - 11[/tex]

    Vertex

    V(-4,-11)

    EXPLANATION

    The given expression is

    [tex]y = {x}^{2} + 8x + 5[/tex]

    We complete the square to get the vertex form.

    Add and subtract half the square of the coefficient of x.

    [tex]y = {x}^{2} + 8x + 16 + 5 - 16[/tex]

    [tex]y = ( {x + 4)}^{2} - 11[/tex]

    The vertex is

    V(-4,-11)

  7. The vertex form of the given equation is f(x) = 2(x-4)²+(-27)  where vertex is (4,-27).

    Step-by-step explanation:

    We have given a quadratic equation in standard form.

    y=  2x²-16x+5

    We have to rewrite given equation in vertex form.

    y  = (x-h)²+k is vertex form of equation where (h,k) is vertex of equation.

    We will use method of completing square to solve this.

    y = 2(x²-8x)+5

    Adding and subtracting  (-4)²  to above equation, we have

    y  = 2(x²-8x+(-4)²)+5-2(-4)²

    y =  2(x-4)²+5-2(16)

    y =  2(x-4)²+ 5 -32

    y =  2(x-4)²-27

    Hence, The vertex form of the given equation is f(x) = 2(x-4)²+(-27)  where vertex is (4,-27).

  8. ANSWER

    Vertex form

    [tex]f(x) = {( x + \frac{7}{2}) }^{2} - \frac{61}{4}[/tex]

    Vertex:

    [tex]V( - \frac{7}{2} , - \frac{6 1}{4} )[/tex]

    EXPLANATION

    The given function is

    [tex]f(x) = {x}^{2} + 7x - 3[/tex]

    Add and subtract the square of half the coefficient of x.

    [tex]f(x) = {x}^{2} + 7x + ( { \frac{7}{2} })^{2} - 3 + ( { \frac{7}{2} })^{2}[/tex]

    The vertex form is

    [tex]f(x) = {( x + \frac{7}{2}) }^{2} - \frac{61}{4}[/tex]

    The vertex is

    [tex]V( - \frac{7}{2} , - \frac{6 1}{4} )[/tex]

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