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Rewrite the equation in vertex form and identify the vertex, the axis of symmetry, and the y-intercept. y=-2x^2+12x-26

Answer:

The equation of the polynomial in vertex form is [tex]y +8= (-2)\cdot (x-3)^{2}[/tex], its vertex is [tex](h,k) = (3, -8)[/tex].

The expression of the axis of symmetry is [tex]x = 3[/tex].

The y-intercept of the function is -26.

Step-by-step explanation:

The vertex form of the second order polynomial is defined by the following expression:

[tex]y-k = C\cdot (x-h)^{2}[/tex]  (1)

Where:

[tex]x[/tex] - Independent variable, dimensionless.

[tex]y[/tex] - Dependent variable, dimensionless.

[tex]h,k[/tex] - Coordinates of the vertex, dimensionless.

[tex]C[/tex] - Vertex constant, dimensionless.

Let [tex]y = -2\cdot x^{2}+12\cdot x - 26[/tex], then we proceed to present the produre for the determination of the vertex form:

1) [tex]y = -2\cdot x^{2}+12\cdot x - 26[/tex] Given

2) [tex]y = (-1)\cdot (2\cdot x^{2})+(-1)\cdot (-12\cdot x) + (-1)\cdot (26)[/tex] [tex](-a)\cdot (-b) = a\cdot b[/tex]/[tex](-a)\cdot b = -a\cdot b[/tex]

3) [tex]y = (-1)\cdot (2\cdot x^{2}-12\cdot x +26)[/tex] Distributive property

4) [tex]y = [(-1)\cdot (2)]\cdot (x^{2}-6\cdot x +13)[/tex] Associative and distributive properties

5) [tex]y = (-2)\cdot [(x^{2}-6\cdot x+9)+4][/tex] [tex](-a)\cdot b = -a\cdot b[/tex]

6) [tex]y = (-2) \cdot [(x-3)^{2}+4][/tex] Perfect square trinomial

7) [tex]y = (-2)\cdot (x-3)^{2}+4\cdot (-2)[/tex] Distributive property

8) [tex]y = (-2)\cdot (x-3)^{2}+(-8)[/tex] [tex](-a)\cdot b = -a\cdot b[/tex]

9) [tex]y +8= (-2)\cdot (x-3)^{2}[/tex] Compatibility of addition/Existence of the additive inverse/Modulative property/Result.

The equation of the polynomial in vertex form is [tex]y +8= (-2)\cdot (x-3)^{2}[/tex], its vertex is [tex](h,k) = (3, -8)[/tex].

The axis of symmetry is a line perpendicular to axis in which the square component of the vertex form is set. The expression of the axis of symmetry is [tex]x = 3[/tex].

The y-intercept is the value of the polynomial when [tex]x = 0[/tex], then, the value is:

[tex]y = -2\cdot (0)^{2}+12\cdot (0) -26[/tex]

[tex]y = -26[/tex]

The y-intercept of the function is -26.

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