The graph of the quadratic function f(x) = x^2 - 6x + c, where c is a constant, has its ve… — SAT

The graph of the quadratic function $f(x) = x^2 - 6x + c$ , where $c$ is a constant, has its vertex at $(h, k)$ in the $xy$ -plane. If the graph of $f$ is translated $3$ units to the right and $2$ units down, the vertex of the translated graph lies on the line $y = 2x$ . What is the value of $c$ ?

Answer: 23

Answer

To find the constant

c

, we first determine the vertex of the function

f(x) = x^2 - 6x + c

. The x-coordinate of the vertex of a parabola in the form

y = ax^2 + bx + c

is given by

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

. For this function,

x = -\frac{-6}{2(1)} = 3

. Substituting this back into the function gives the y-coordinate of the vertex:

f(3) = (3)^2 - 6(3) + c = c - 9

. Thus, the original vertex is at

(3, c - 9)

. Translating the graph

3

units to the right increases the x-coordinate of the vertex by

3

, making it

3 + 3 = 6

. Translating the graph

2

units down decreases the y-coordinate of the vertex by

2

, making it

(c - 9) - 2 = c - 11

. The problem states that this new vertex

(6, c - 11)

lies on the line

y = 2x

. Substituting these coordinates into the linear equation gives

c - 11 = 2(6)

, which simplifies to

c - 11 = 12

. Adding

11

to both sides gives the value of

c

23

Step-by-Step Solution

Find the vertex

(h, k)

of the original quadratic function

f(x) = x^2 - 6x + c

The vertex is at

(3, c - 9)

The x-coordinate of the vertex of a quadratic function

y = ax^2 + bx + c

is given by

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

. For

f(x) = x^2 - 6x + c

, we have

h = -\frac{-6}{2(1)} = 3

. Substituting

x = 3

into the function gives the y-coordinate:

k = f(3) = 3^2 - 6(3) + c = c - 9

Determine the coordinates of the vertex after translating the graph

3

units to the right and

2

units down.

The new vertex is at

(6, c - 11)

A translation of

3

units to the right increases the x-coordinate of the vertex by

3

, so the new x-coordinate is

3 + 3 = 6

. A translation of

2

units down decreases the y-coordinate of the vertex by

2

, so the new y-coordinate is

(c - 9) - 2 = c - 11

Set up an equation using the line

y = 2x

and solve for

c

c = 23

Since the translated vertex

(6, c - 11)

lies on the line

y = 2x

, substituting

x = 6

and

y = c - 11

into the line's equation must satisfy it:

c - 11 = 2(6) \Rightarrow c - 11 = 12 \Rightarrow c = 23

Key Concept

Determining the vertex of a quadratic function and applying translations to its graph.

Estimated Time:1m 30s

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