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A function is shown below where b is a real number.
f(x) = x2 + bx + 182
The minimum of the function is 13.
Create an equivalent equation of the function in the form f(x) = (x - h)2 + k.
Type your numerical answers below for h and k. Use the hyphen (-) for the negative sign if
necessary.

Respuesta :

The equivalent quadratic function in vertex form is:

y = (x + 12)² - 13.

What is the vertex of a quadratic equation?

A quadratic equation is modeled by:

y = ax^2 + bx + c

The vertex is given by:

[tex](x_v, y_v)[/tex]

In which:

  • [tex]x_v = -\frac{b}{2a}[/tex]
  • [tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]

Considering the coefficient a, we have that:

  • If a < 0, the vertex is a maximum point.
  • If a > 0, the vertex is a minimum point.

In this problem, the function is:

f(x) = x² + bx + 182.

The coefficients are of a = 1 and c = 182. The minimum is of [tex]y_v = 13[/tex], hence we use it to solve for b.

[tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]

[tex]13 = -\frac{b^2 - 4(1)(182)}{4}[/tex]

b² - 728 = -52

b² = 676

b = sqrt(676)

b = 24.

The x-value of the vertex is:

[tex]x_v = -\frac{b}{2a} = -\frac{24}{2} = -12[/tex]

What is the equation of a parabola given it’s vertex?

The equation of a quadratic function, of vertex (h,k), is given by:

y = a(x - h)² + k

In which a is the leading coefficient.

For this problem, we have that a = 1, h = -12 and k = 13, hence:

y = (x + 12)² - 13.

More can be learned about quadratic functions at https://brainly.com/question/24737967

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