A parabola in the standard (x, y) coordinate plane is defined by the equation y = \frac{1}… — ACT

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $y = \frac{1}{12}x^2 - 2x + 13$ . What is the $y$ -coordinate of the focus of this parabola?

Cevap: 4

Cevap

The y-coordinate of the focus is 4.

The standard form of the parabola is

y - 1 = \frac{1}{12}(x - 12)^2

. Comparing this to

y - k = \frac{1}{4p}(x - h)^2

gives the vertex

(h, k) = (12, 1)

and

4p = 12 \implies p = 3

. Since the parabola opens upward, the focus is at

(12, 1 + 3) = (12, 4)

, making the

y

-coordinate 4.

Adım Adım Çözüm

Complete the square to rewrite the equation in standard vertex form.

y = \frac{1}{12}(x - 12)^2 + 1

Rewriting the general quadratic equation into standard form allows us to directly identify the vertex and focal parameters.

Equate the coefficients to find the focal distance

p

and vertex

(h, k)

Vertex

(h, k) = (12, 1)

and

p = 3

The standard vertex form of a vertical parabola is

y - k = \frac{1}{4p}(x - h)^2

. Setting

\frac{1}{4p} = \frac{1}{12}

gives

p = 3

Calculate the focus coordinates

(h, k + p)

Focus

= (12, 4)

, so the

y

-coordinate is

4

For an upward-opening parabola, the focus is located

p

units directly above the vertex.

Anahtar Kavram

Rewriting a quadratic equation into standard vertex form to find the properties of a parabola, including its vertex and focus.

Soru

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