A parabola in the xy-plane has equation y = a(x - h)^2 + k, where a, h, and k are constant… — SAT

A parabola in the $xy$ -plane has equation $y = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The parabola passes through the points $(2, 13)$ and $(8, 13)$ in the $xy$ -plane. If the minimum value of the quadratic function defined by this equation is $-5$ , what is the value of $a$ ?

Answer: 2

Answer

The correct answer is 2.

The correct answer is 2. The axis of symmetry of the parabola is halfway between the x-coordinates of the two symmetric points

(2, 13)

and

(8, 13)

, which is

x = 5

. Since the minimum value of the function is

-5

, the vertex of the parabola is

(5, -5)

. Writing the equation in vertex form,

y = a(x - 5)^2 - 5

, and substituting the point

(2, 13)

yields

13 = a(2 - 5)^2 - 5

, which simplifies to

18 = 9a

, so

a = 2

Step-by-Step Solution

Find the axis of symmetry

x = h

using the symmetric points

(2, 13)

and

(8, 13)

h = 5

Since the points

(2, 13)

and

(8, 13)

have the same

y

-coordinate, the axis of symmetry of the parabola must lie exactly halfway between their

x

-coordinates:

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

Determine the vertex

(h, k)

of the parabola.

(5, -5)

The minimum value of a quadratic function occurs at its vertex. The given minimum value is

-5

, which represents the

y

-coordinate (

k

) of the vertex. Combining this with

h = 5

, the vertex is

(5, -5)

Write the equation of the parabola in vertex form.

y = a(x - 5)^2 - 5

The vertex form of a quadratic equation is

y = a(x - h)^2 + k

. Substituting the values

h = 5

and

k = -5

yields this equation.

Substitute one of the given points into the equation to solve for

a

a = 2

Substituting

(2, 13)

into the equation gives

13 = a(2 - 5)^2 - 5 \Rightarrow 18 = 9a \Rightarrow a = 2

Key Concept

Finding the equation of a parabola using symmetry and vertex form

Estimated Time:2m 0s

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