In the xy-plane, the graph of the quadratic function f(x) = x^2 - 4x - 5 intersects the x-… — SAT

In the $xy$ -plane, the graph of the quadratic function $f(x) = x^2 - 4x - 5$ intersects the $x$ -axis at the points $(p, 0)$ and $(q, 0)$ and has vertex $(h, k)$ . What is the area of the triangle with vertices at $(p, 0)$ , $(q, 0)$ , and $(h, k)$ ?

Answer: 27

Answer

The area of the triangle is 27.

To find the area of the triangle, we first determine the coordinates of its vertices. The base of the triangle lies on the

x

-axis, with endpoints at the

x

-intercepts of the function

f(x) = x^2 - 4x - 5

. Solving

x^2 - 4x - 5 = 0

by factoring gives

(x - 5)(x + 1) = 0

, so the intercepts are at

x = -1

and

x = 5

. The distance between these two points is

5 - (-1) = 6

, which is the base of the triangle. The third vertex is the vertex of the parabola. The

x

-coordinate of the vertex is

h = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2

. Substituting

x = 2

into the function gives the

y

-coordinate:

k = f(2) = 2^2 - 4(2) - 5 = -9

. The height of the triangle is the vertical distance from the

x

-axis to the vertex, which is

|-9| = 9

. The area of the triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

Step-by-Step Solution

Find the

x

-intercepts of the parabola.

The intercepts are

(-1, 0)

and

(5, 0)

Setting

f(x) = 0

gives

x^2 - 4x - 5 = 0

. Factoring the quadratic equation yields

(x - 5)(x + 1) = 0

, which gives

x = 5

and

x = -1

Calculate the base of the triangle.

The base length is

6

The base of the triangle is the segment along the

x

-axis between the two intercepts. The distance between

(-1, 0)

and

(5, 0)

5 - (-1) = 6

Find the vertex

(h, k)

of the parabola.

The vertex is at

(2, -9)

The

x

-coordinate of the vertex is the midpoint of the intercepts:

h = \frac{-1 + 5}{2} = 2

. The

y

-coordinate is

k = f(2) = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9

Calculate the area of the triangle.

The area is

27

The height of the triangle is the distance from the

x

-axis to the vertex, which is

|k| = |-9| = 9

. Using the formula for the area of a triangle,

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

Key Concept

Finding the

x

-intercepts and vertex of a quadratic function to solve geometric problems in the coordinate plane.

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