A parabola in the xy-plane has vertex (2, 11) and passes through the point (5, -7). The eq… — SAT

A parabola in the $xy$ -plane has vertex $(2, 11)$ and passes through the point $(5, -7)$ . The equation of the parabola is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Answer: 9

Answer

The value of

a + b + c

9

The correct value of

a + b + c

9

. This is found by writing the parabola's equation in vertex form as

y = -2(x - 2)^2 + 11

and expanding it to standard form

y = -2x^2 + 8x + 3

, which gives the coefficients

a = -2

b = 8

, and

c = 3

. Alternatively, substituting

x = 1

directly into the vertex form gives

f(1) = a(1)^2 + b(1) + c = -2(1 - 2)^2 + 11 = 9

Step-by-Step Solution

Write the equation of the parabola in vertex form using the given vertex

(2, 11)

y = a(x - 2)^2 + 11

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex of the parabola.

Substitute the coordinates of the point

(5, -7)

into the vertex form equation to solve for the constant

a

a = -2

Since the point

(5, -7)

lies on the parabola, substituting

x = 5

and

y = -7

allows us to solve the linear equation

-7 = a(5 - 2)^2 + 11

for

a

Expand the vertex form equation

y = -2(x - 2)^2 + 11

into standard form

y = ax^2 + bx + c

to identify the coefficients

a

b

, and

c

y = -2x^2 + 8x + 3

, which gives

a = -2

b = 8

, and

c = 3

Expanding the squared term and distributing the coefficient

-2

converts the equation to standard form, making it easy to read off the coefficients.

Calculate the sum of the coefficients

a + b + c

a + b + c = 9

Adding the identified coefficients:

-2 + 8 + 3 = 9

Key Concept

Writing and converting quadratic functions between vertex form

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

and standard form

y = ax^2 + bx + c

Alternative Method

Instead of expanding the vertex form equation to find the individual coefficients

a

b

, and

c

, recognize that the expression

a + b + c

is equal to

f(1)

for the quadratic function

f(x) = ax^2 + bx + c

. After finding

a = -2

using the vertex form

f(x) = a(x - 2)^2 + 11

, directly evaluate

f(1) = -2(1 - 2)^2 + 11 = 9

Estimated Time:2m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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