The function f is defined by f(x) = 2x^2 - 12x + c, where c is a constant. In the xy-plane… — SAT

The function $f$ is defined by $f(x) = 2x^2 - 12x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(h, 5)$ , where $h$ is a constant. What is the value of $c$ ?

Answer: 23

Answer

For a quadratic function in standard form

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

, the x-coordinate of the vertex is given by

h = -\frac{b}{2a}

. In this function,

a = 2

and

b = -12

, so

h = -\frac{-12}{2(2)} = 3

. The vertex coordinates are

(3, 5)

, which means

f(3) = 5

. Substituting

x = 3

into the function gives

2(3)^2 - 12(3) + c = 5

. Simplifying the terms gives

18 - 36 + c = 5

, which becomes

-18 + c = 5

. Adding

18

to both sides yields

c = 23

Step-by-Step Solution

Identify the x-coordinate formula for the vertex of a quadratic function

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

The x-coordinate

h

is given by

h = -\frac{b}{2a}

To find the axis of symmetry and the horizontal position of the vertex.

Substitute

a = 2

and

b = -12

from the given equation

f(x) = 2x^2 - 12x + c

into the vertex formula.

h = -\frac{-12}{2(2)} = 3

To calculate the specific x-coordinate of the vertex for this function.

Substitute the vertex coordinates

(3, 5)

into the function

f(x)

f(3) = 2(3)^2 - 12(3) + c = 5

Since the vertex lies on the graph of the function, its coordinates must satisfy the function's equation.

Simplify the equation and solve for the constant

c

18 - 36 + c = 5 \implies -18 + c = 5 \implies c = 23

To determine the final value of the constant

c

Key Concept

Vertex of a quadratic function

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