Math Problem Statement

Graph the following function. Then use geometry​ (not Riemann​ sums) to find the area and the net area of the region described. The region between the graph of yequals=StartAbsoluteValue x EndAbsoluteValue minus 2x−2 and the​ x-axis for negative 3 less than or equals x less than or equals 3−3≤x≤3. Question content area bottom Part 1 Select the correct graph below. A. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A V-shaped graph opens downward from a vertex at (0, 2) and passes through (negative 1, 1) and (1, 1). B. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A V-shaped graph opens upward from a vertex at (0, negative 2) and passes through (negative 1, negative 1) and (1, negative 1). Your answer is correct.C. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A V-shaped graph opens downward from a vertex at (0, negative 2) and passes through (negative 1, negative 3) and (1, negative 3). D. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A V-shaped graph opens upward from a vertex at (0, 2) and passes through (negative 1, 3) and (1, 3). Part 2 The area of the region is enter your response here. ​(Simplify your​ answer.)

Solution

To solve this, let's break it down into two parts:

Part 1: Identify the Correct Graph

The function given is: y=∣x∣−2y = |x| - 2 This is a V-shaped function due to the absolute value ∣x∣|x|, shifted down by 2 units because of the −2-2.

  1. The absolute value function ∣x∣|x| typically has a vertex at the origin (0, 0) and opens upward.
  2. Subtracting 2 from the absolute value moves the vertex down by 2 units to the point (0, -2).

Thus, the function should have:

  • A vertex at (0, -2),
  • The shape of a V opening upwards from this vertex.

Looking at the choices:

  • Option B: This graph shows a V-shaped function opening upward with the vertex at (0, -2), passing through points like (-1, -1) and (1, -1), which matches our expected shape and transformation.

Therefore, the correct answer is Option B.

Part 2: Find the Area of the Region Using Geometry

We are tasked to find the area between this function and the x-axis over the interval [−3,3][-3, 3].

  1. Determine the intersection points with the x-axis:

    • Setting y=0y = 0 in y=∣x∣−2y = |x| - 2, ∣x∣−2=0⇒∣x∣=2|x| - 2 = 0 \Rightarrow |x| = 2 So, x=2x = 2 and x=−2x = -2.
    • This gives us two segments from [−3,−2][-3, -2], [−2,2][-2, 2], and [2,3][2, 3].

  2. Divide the region into two triangles and a rectangle:

    • The V-shape creates two triangular regions from x=−2x = -2 to x=2x = 2, and two triangular regions from [−3,−2][-3, -2] and [2,3][2, 3] touching the x-axis.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Geometry
Area Calculation

Formulas

Area of a triangle = (1/2) * base * height

Theorems

Properties of absolute value functions

Suitable Grade Level

Grades 9-12

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