A quadratic function f is defined by f(x) = a(x - 4)(x - 10), where a is a positive consta… — SAT

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - 10)$ , where $a$ is a positive constant. In the $xy$ -plane, the graph of $f$ has a vertex with a $y$ -coordinate of $-18$ . What is the value of $a$ ?

Answer: 2

Answer

The value of

a

is 2.

The vertex of a parabola lies on the axis of symmetry, which is located midway between its x-intercepts. Since the function is

f(x) = a(x - 4)(x - 10)

, the x-intercepts are at

x = 4

and

x = 10

. The midpoint of these values is

\frac{4 + 10}{2} = 7

. Thus, the x-coordinate of the vertex is 7. Since the y-coordinate of the vertex is given as

-18

, the vertex is at

(7, -18)

. Substituting these coordinates into the function equation yields

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = a(3)(-3)

, or

-18 = -9a

. Solving for

a

gives

a = 2

Step-by-Step Solution

Find the x-coordinate of the vertex using the symmetry of the parabola.

The x-coordinate of the vertex is 7.

The x-coordinate of the vertex lies on the axis of symmetry, which is the midpoint of the x-intercepts

x = 4

and

x = 10

Identify the coordinates of the vertex.

The vertex is at

(7, -18)

The y-coordinate of the vertex is given as

-18

, and its x-coordinate was calculated as 7.

Substitute the vertex coordinates into the function definition to solve for

a

a = 2

Substituting

(7, -18)

into

f(x) = a(x - 4)(x - 10)

gives

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = -9a

Key Concept

Finding the vertex of a quadratic function from its factored form and solving for a leading coefficient.

Alternative Method

Alternatively, the function

f(x) = a(x - 4)(x - 10)

can be expanded to standard form:

f(x) = a(x^2 - 14x + 40)

. The x-coordinate of the vertex can be found using the formula

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

, which gives

x = -\frac{-14a}{2a} = 7

. Then, substitute

x = 7

and

f(7) = -18

to solve for

a

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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