To determine the solution region of the system of inequalities:
[tex]\[
\begin{array}{l}
y \geq 3(0.8)^x \\
y \geq x^2-5
\end{array}
\][/tex]
we must follow these steps:
### Step 1: Graph the Equations
We start by graphing the equations [tex]\( y = 3(0.8)^x \)[/tex] and [tex]\( y = x^2 - 5 \)[/tex]. These curves will help us identify the boundaries of the solution region.
1. Graph [tex]\( y = 3(0.8)^x \)[/tex]:
- This is an exponential decay function because [tex]\(0.8 < 1\)[/tex]. As [tex]\(x\)[/tex] increases, [tex]\(3(0.8)^x\)[/tex] decreases.
- When [tex]\(x = 0\)[/tex], [tex]\(y = 3\)[/tex].
- As [tex]\(x\)[/tex] tends to [tex]\(+\infty\)[/tex], [tex]\(3(0.8)^x\)[/tex] tends towards 0.
- As [tex]\(x\)[/tex] tends to [tex]\(-\infty\)[/tex], [tex]\(3(0.8)^x\)[/tex] grows exponentially.
2. Graph [tex]\( y = x^2 - 5 \)[/tex]:
- This is a parabola that opens upwards.
- The vertex is at [tex]\((0, -5)\)[/tex].
### Step 2: Determine the Inequality Regions
Next, we determine where the inequalities hold.
1. Region for [tex]\( y \geq 3(0.8)^x \)[/tex]:
- This region lies above or on the curve [tex]\( y = 3(0.8)^x \)[/tex].
2. Region for [tex]\( y \geq x^2 - 5 \)[/tex]:
- This region lies above or on the parabola [tex]\( y = x^2 - 5 \)[/tex].
### Step 3: Find the Intersection Points
The intersection points of the two curves indicate where we will further analyze regions:
- Solve for [tex]\(3(0.8)^x = x^2 - 5\)[/tex]. This involves finding the values of [tex]\(x\)[/tex] where these two curves intersect.
### Step 4: Determine the Solution Region
Identify the overlapping region:
- Since both [tex]\( y \geq 3(0.8)^x \)[/tex] and [tex]\( y \geq x^2 - 5 \)[/tex] must be true for the entire region:
- The solution region lies above both curves.
### Step 5: Illustrate on the Graph
Let's illustrate:
1. Plot [tex]\( y = 3(0.8)^x \)[/tex].
2. Plot [tex]\( y = x^2 - 5 \)[/tex].
3. Shade the region above the first curve.
4. Shade the region above the second curve.
5. Identify the common shaded area.
### Final Answer
The correct graph will display both functions and highlight the region where both inequalities are satisfied simultaneously. The shaded solution region should show above both curves in the upper part of the graph.
