In the xy-plane, the vertex of the parabola defined by y = a(x - 2)(x - 8) has a y-coordin… — SAT

In the $xy$ -plane, the vertex of the parabola defined by $y = a(x - 2)(x - 8)$ has a $y$ -coordinate of $-18$ , where $a$ is a positive constant. What is the value of $a$ ?

Answer: 2

Answer

The correct answer is 2. The quadratic function is given in factored form as

y = a(x - 2)(x - 8)

. The x-intercepts of this parabola are at

x = 2

and

x = 8

. Because of the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts:

x = \frac{2 + 8}{2} = 5

. The y-coordinate of the vertex is given as

-18

, meaning the vertex is at the point

(5, -18)

. Substituting these coordinates into the equation gives

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3) = -9a

. Solving for

a

yields

a = 2

Step-by-Step Solution

Identify the x-intercepts from the factored form equation

y = a(x - 2)(x - 8)

and find the x-coordinate of the vertex.

The x-intercepts are

x = 2

and

x = 8

. The x-coordinate of the vertex is the midpoint of the intercepts:

x = \frac{2 + 8}{2} = 5

The axis of symmetry of a parabola passes through its vertex and lies midway between its x-intercepts.

Substitute the coordinates of the vertex

(5, -18)

into the quadratic equation to solve for the constant

a

Substituting

x = 5

and

y = -18

yields

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3)

, so

-18 = -9a

, giving

a = 2

Since the vertex is a point on the parabola, its coordinates must satisfy the equation of the parabola.

Key Concept

Using the symmetry of quadratic functions in factored form to find the vertex coordinates.

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