The shaded region shown below is bounded by the functions f, of, x, equals, minus, 2, x, squared, plus, 0, point, 7, 5, x, plus, 6f(x)=−2x 
2
 +0.75x+6 and g, of, x, equals, minus, 0, point, 5, x, plus, 4g(x)=−0.5x+4 and the xx and yy axes. Find the area of the shaded region using a calculator. Round your answer to the nearest thousandth.
x

To find the area of the shaded region bounded by the functions \( f(x) = -2x^2 + 0.75x + 6 \), \( g(x) = -0.5x + 4 \), and the x- and y-axes, we need to follow these steps:

### 1. **Determine the points of intersection:**
We first need to find where the functions \( f(x) \) and \( g(x) \) intersect each other, as well as where each function intersects the x-axis (where \( f(x) = 0 \) and \( g(x) = 0 \)).

#### a. **Intersection of \( f(x) \) with the x-axis:**
Set \( f(x) = 0 \):
\[
-2x^2 + 0.75x + 6 = 0
\]
This is a quadratic equation. We will use the quadratic formula to find the values of \( x \).

#### b. **Intersection of \( g(x) \) with the x-axis:**
Set \( g(x) = 0 \):
\[
-0.5x + 4 = 0
\]
Solving for \( x \):
\[
x = \frac{4}{0.5} = 8
\]

#### c. **Intersection of \( f(x) \) and \( g(x) \):**
Set \( f(x) = g(x) \):
\[
-2x^2 + 0.75x + 6 = -0.5x + 4
\]
Solve for \( x \) to find the point of intersection.

### 2. **Set up the integral for the area:**
After finding the limits of integration, the total area will be the integral of the difference between the two functions over the appropriate interval. The formula will look like this:
\[
\text{Area} = \int_{x_1}^{x_2} |f(x) - g(x)| \, dx
\]
where \( x_1 \) and \( x_2 \) are the intersection points.

### 3. **Use a calculator to evaluate the integrals.**
Once we have the correct limits of integration and the expressions for \( f(x) \) and \( g(x) \), we can calculate the area.

Let me quickly do the calculations for the intersections and the area. sympy as sp

# Define the variables and functions
x = sp.symbols('x')
f_x = -2*x**2 + 0.75*x + 6
g_x = -0.5*x + 4

# Solve for the x-intercepts of f(x) and g(x)
f_x_intercepts = sp.solve(f_x, x)
g_x_intercepts = sp.solve(g_x, x)

# Solve for the intersection of f(x) and g(x)
intersection_points = sp.solve(f_x - g_x, x)

# Now we integrate between the limits of integration
# Let's assume the limits are from x = 0 to the x-coordinate where the two functions intersect.
area = sp.integrate(f_x - g_x, (x, 0, intersection_points[0]))

# Evaluate the solutions
f_x_intercepts, g_x_intercepts, intersection_points, areaHere are the results:

1. **x-intercepts:**
   - For \( f(x) = 0 \), the x-intercepts are approximately \( x = -1.555 \) and \( x = 1.930 \).
   - For \( g(x) = 0 \), the x-intercept is \( x = 8 \).

2. **Intersection of \( f(x) \) and \( g(x) \):**
   The functions \( f(x) \) and \( g(x) \) intersect at approximately \( x = -0.735 \) and \( x = 1.360 \).

### Area Calculation:
The integral yields a value of approximately **-0.868**. However, since area is always positive, we take the absolute value of this result.

Thus, the area of the shaded region is approximately **0.868 square units** (rounded to the nearest thousandth).

Would you like more details on any part of this process? Here are some questions that might help:

1. How do we find the area between two curves using integration?
2. Can you explain the quadratic formula and how it is used to find intercepts?
3. How do we handle negative values when calculating area?
4. What is the significance of the intersection points between the two functions?
5. How does the definite integral relate to the geometric area under a curve?

**Tip:** When solving for the area under curves, always double-check the limits of integration and ensure that the difference between the functions is positive.

The shaded region shown below is bounded by the functions f(x) = -2x^2 + 0.75x + 6 and g(x) = -0.5x + 4 and the x and y axes. Find the area of the shaded region using a calculator. Round your answer to the nearest thousandth.

Learn how to calculate the area between the quadratic function f(x) = -2x^2 + 0.75x + 6 and the linear function g(x) = -0.5x + 4, bounded by the x and y axes. The problem is solved using integration, and the area is found to be approximately 0.868 square units. This solution is suitable for advanced high school students in Grades 11-12.

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