What is the total area of the shaded regions? **Understanding the Quadratic Function: $ f(x) = 16 - x^2 $**The graph represents the quadratic function $ f(x) = 16 - x^2 $, which is a downward-opening parabola.**Graph Description:**- **Parabola:** The curve is symmetric around the y-axis, reaching its maximum point at $ y = 16 $ (the vertex), located at the origin where $ x = 0 $.- **Intercepts:** - **X-intercepts:** The parabola intersects the x-axis at $ x = -4 $ and $ x = 4 $. - **Y-intercept:** The curve intersects the y-axis at $ y = 16 $. **Axes and Range:**- **X-axis:** The horizontal axis represents the input values of the function, $ x $, ranging from -6 to 6.- **Y-axis:** The vertical axis represents the output values of the function, $ y $, ranging up to 16. **Graph Features:**- **Shaded Area:** The shaded region under the curve between $ x = -4 $ and $ x = 4 $ highlights the area bounded by the function and the x-axis.- **Grid Lines:** Supportive grid lines are present to assist in measuring and depicting the function's values and symmetry. **Properties:**- **Domain:** All real numbers ($ -\infty, \infty $).- **Range:** $ y \leq 16 $, because the maximum value is at the vertex height.This visualization aids in understanding how the algebraic expression of the quadratic function corresponds with its graphical representation. Understanding these elements helps interpret the behavior of quadratic functions in mathematical and real-world contexts.

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Math

Calculus

What is the total area of the shaded regions

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 58E

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Question

What is the total area of the shaded regions?

**Understanding the Quadratic Function: $ f(x) = 16 - x^2 $**

The graph represents the quadratic function $ f(x) = 16 - x^2 $, which is a downward-opening parabola.

**Graph Description:**

- **Parabola:** The curve is symmetric around the y-axis, reaching its maximum point at $ y = 16 $ (the vertex), located at the origin where $ x = 0 $.
- **Intercepts:**
- **X-intercepts:** The parabola intersects the x-axis at $ x = -4 $ and $ x = 4 $.
- **Y-intercept:** The curve intersects the y-axis at $ y = 16 $.

**Axes and Range:**

- **X-axis:** The horizontal axis represents the input values of the function, $ x $, ranging from -6 to 6.
- **Y-axis:** The vertical axis represents the output values of the function, $ y $, ranging up to 16.

**Graph Features:**

- **Shaded Area:** The shaded region under the curve between $ x = -4 $ and $ x = 4 $ highlights the area bounded by the function and the x-axis.
- **Grid Lines:** Supportive grid lines are present to assist in measuring and depicting the function's values and symmetry.

**Properties:**

- **Domain:** All real numbers ($ -\infty, \infty $).
- **Range:** $ y \leq 16 $, because the maximum value is at the vertex height.

This visualization aids in understanding how the algebraic expression of the quadratic function corresponds with its graphical representation. Understanding these elements helps interpret the behavior of quadratic functions in mathematical and real-world contexts.

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