To find the solution region for the system of inequalities:
[tex]\[
\begin{array}{l}
y \geq 2(1.3)^x \\
y \leq -x^2 + 4x + 5
\end{array}
\][/tex]
we need to analyze and graph each inequality step-by-step.
### Step 1: Graph the Inequality [tex]\(y \geq 2(1.3)^x\)[/tex]
1. Understand the Expression: [tex]\(y = 2(1.3)^x\)[/tex] is an exponential function.
2. Plot the Function: Find a few points to understand the shape. For instance:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 2(1.3)^0 = 2\)[/tex].
- When [tex]\(x = 1\)[/tex], [tex]\(y = 2(1.3)^1 = 2.6\)[/tex].
- When [tex]\(x = -1\)[/tex], [tex]\(y = 2(1.3)^{-1} \approx 1.54\)[/tex].
The exponential function rises as [tex]\(x\)[/tex] increases and approaches 0 as [tex]\(x\)[/tex] decreases.
3. Shade the Region: Because the inequality is [tex]\(y \geq 2(1.3)^x\)[/tex], shade the region above the curve.
### Step 2: Graph the Inequality [tex]\(y \leq -x^2 + 4x + 5\)[/tex]
1. Understand the Expression: [tex]\(y = -x^2 + 4x + 5\)[/tex] is a quadratic function opening downwards.
2. Find Key Points: Identify the vertex and the intercepts.
- The vertex can be found using [tex]\(x = -\frac{b}{2a}\)[/tex]. Here, [tex]\(a = -1\)[/tex], [tex]\(b = 4\)[/tex]. So [tex]\(x = -\frac{4}{2(-1)} = 2\)[/tex]. Substitute [tex]\(x = 2\)[/tex] back into the equation to get [tex]\(y = -2^2 + 4(2) + 5 = 9\)[/tex].
- The y-intercept is at [tex]\(y = 5\)[/tex] when [tex]\(x = 0\)[/tex].
- Find the x-intercepts by solving [tex]\( -x^2 + 4x + 5 = 0 \)[/tex]:
[tex]\[
x^2 - 4x - 5 = 0 \implies (x-5)(x+1) = 0 \implies x = 5 \text{ or } x = -1.
\][/tex]
3. Plot the Quadratic Function: Draw a parabola opening downwards with a vertex at [tex]\((2, 9)\)[/tex], passing through [tex]\((0, 5)\)[/tex], [tex]\((5, 0)\)[/tex], and [tex]\((-1, 0)\)[/tex].
4. Shade the Region: Because the inequality is [tex]\(y \leq -x^2 + 4x + 5\)[/tex], shade the region below the curve.
### Step 3: Find the Intersection of the Shaded Regions
The solution to the system of inequalities is the region where the shaded areas overlap.
### Analyzing the Graphs:
- Ensure that the exponential function and the parabola intersect.
- The correct graph should have the shaded region above the exponential curve [tex]\(y = 2(1.3)^x\)[/tex] and below the quadratic curve [tex]\(y = -x^2 + 4x + 5\)[/tex].
Since actual graphs are not provided in the question, the correct answer should reflect these conditions. This means you'll typically be looking for:
- Both curves accurately drawn.
- The intersection points are shown clearly.
- The shaded region respects the inequalities.
If you are selecting from provided options, choose the graph that correctly meets these criteria based on the intersection and shading described.
