The graph of the quadratic equation y = 2x^2 - 12x + c, where c is a constant, is a parabo… — SAT

The graph of the quadratic equation $y = 2x^2 - 12x + c$ , where $c$ is a constant, is a parabola in the $xy$ -plane. If the $y$ -coordinate of the vertex of this parabola is $-5$ , what is the value of $c$ ?

Answer: 13

Answer

To find the value of the constant

c

, we calculate the coordinates of the vertex of the parabola. The x-coordinate of the vertex for a quadratic function in standard form

y = ax^2 + bx + c

is given by

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

. Substituting

a = 2

and

b = -12

gives

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

. Evaluating the quadratic equation at

x = 3

gives the y-coordinate of the vertex:

y = 2(3)^2 - 12(3) + c = 18 - 36 + c = c - 18

. Since we are given that the y-coordinate of the vertex is

-5

, we set

c - 18 = -5

and solve to find

c = 13

Step-by-Step Solution

Calculate the x-coordinate of the vertex using the vertex formula.

x = 3

The axis of symmetry and the x-coordinate of the vertex are located at

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

Substitute the x-coordinate of the vertex into the equation to express the y-coordinate in terms of

c

y = c - 18

Evaluating the quadratic function at the vertex's x-coordinate gives the minimum or maximum value of the function.

Equate the expression for the y-coordinate to the given vertex y-coordinate of

-5

and solve for

c

c = 13

Setting the calculated y-coordinate expression equal to the given value allows us to isolate and solve for the unknown constant.

Key Concept

Determining the vertex of a quadratic function from its standard form and solving for a constant coefficient.

Alternative Method

Alternatively, we can complete the square to write the quadratic equation in vertex form,

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

. Factoring the leading coefficient from the variable terms gives

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

. To complete the square inside the parentheses, add and subtract

9

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

. In this vertex form, the y-coordinate of the vertex is

k = c - 18

. Since the vertex y-coordinate is

-5

, we set

c - 18 = -5

to get

c = 13

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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