A parabola in the xy-plane has vertex (3, 18) and passes through the origin. If the equati… — SAT

A parabola in the $xy$ -plane has vertex $(3, 18)$ and passes through the origin. If the equation of the parabola is written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b$ ?

Answer: 10

Answer

The vertex form of a quadratic function with vertex

(h, k)

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

. Substituting the given vertex

(3, 18)

yields the equation

y = a(x - 3)^2 + 18

. Since the parabola passes through the origin, we can substitute the point

(0, 0)

into the equation to find the value of

a

0 = a(0 - 3)^2 + 18 \implies 9a = -18 \implies a = -2

. Substituting

a = -2

back into the vertex form and expanding gives

y = -2(x - 3)^2 + 18 = -2(x^2 - 6x + 9) + 18 = -2x^2 + 12x

. Comparing this to the standard form

y = ax^2 + bx + c

, we identify

a = -2

and

b = 12

. The sum of these constants is

a + b = -2 + 12 = 10

Step-by-Step Solution

Write the equation of the parabola in vertex form.

y = a(x - 3)^2 + 18

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

Substitute the coordinates of the origin

(0, 0)

to solve for

a

a = -2

The parabola passes through the origin

(0, 0)

, so these coordinates must satisfy the equation.

Expand the vertex form equation into standard form

y = ax^2 + bx + c

y = -2x^2 + 12x

Expanding the equation allows us to identify the coefficients

a

and

b

directly.

Identify the values of

a

and

b

, and calculate

a + b

a + b = 10

Comparing

y = -2x^2 + 12x

y = ax^2 + bx + c

gives

a = -2

and

b = 12

, so

a + b = -2 + 12 = 10

Key Concept

Vertex form of quadratic functions and conversion to standard form

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