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Question

$$ 1<\quad $$ (a) Graph the following quadratic function on $$ [-10,10] $$ by $$ [-10,10] $$ $$ g(x)=-5 x^{2}-2 x+8 $$ (b) Does this window give a corplete graph?

UpStudy AI · Accepted Answer

Sure, let's tackle both parts of your question step by step.

---

### **(a) Graphing the Quadratic Function $$ g(x) = -5x^{2} - 2x + 8 $$ on the Window $$[-10, 10] \times [-10, 10]$$**

Since I can't provide a visual graph here, I'll describe how the graph of the quadratic function $$ g(x) = -5x^{2} - 2x + 8 $$ would appear within the specified window.

#### **Key Characteristics of the Function:**

1. **Direction of Opening:**
   - The coefficient of $$ x^{2} $$ is **-5**, which is negative. This means the parabola opens **downwards**.

2. **Vertex (Maximum Point):**
   - The vertex form of a parabola $$ ax^{2} + bx + c $$ is found using:
     $$
     x_v = -\frac{b}{2a}
     $$
     $$
     y_v = g(x_v)
     $$
     
   - Calculating $$ x_v $$:
     $$
     x_v = -\frac{-2}{2 \times -5} = -\frac{2}{-10} = 0.2
     $$
     
   - Calculating $$ y_v $$:
     $$
     g(0.2) = -5(0.2)^2 - 2(0.2) + 8 = -5(0.04) - 0.4 + 8 = -0.2 - 0.4 + 8 = 7.4
     $$
     
   - **Vertex:** $$ (0.2, 7.4) $$

3. **Y-Intercept:**
   - Set $$ x = 0 $$:
     $$
     g(0) = -5(0)^2 - 2(0) + 8 = 8
     $$
   - **Y-Intercept:** $$ (0, 8) $$

4. **X-Intercepts (Roots):**
   - Solve $$ -5x^{2} - 2x + 8 = 0 $$:
     $$
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{4 - 4(-5)(8)}}{2(-5)} = \frac{2 \pm \sqrt{4 + 160}}{-10} = \frac{2 \pm \sqrt{164}}{-10}
     $$
     $$
     \sqrt{164} \approx 12.81
     $$
     $$
     x \approx \frac{2 \pm 12.81}{-10}
     $$
     $$
     x \approx \frac{2 + 12.81}{-10} \approx -1.48 \quad \text{and} \quad x \approx \frac{2 - 12.81}{-10} \approx 1.08
     $$
   - **X-Intercepts:** Approximately $$ (-1.48, 0) $$ and $$ (1.08, 0) $$

5. **Behavior at Window Boundaries:**
   - **At $$ x = 10 $$:**
     $$
     g(10) = -5(100) - 2(10) + 8 = -500 - 20 + 8 = -512
     $$
     - Within the y-window $$[-10, 10]$$, $$ y = -512 $$ is **not visible**.
     
   - **At $$ x = -10 $$:**
     $$
     g(-10) = -5(100) - 2(-10) + 8 = -500 + 20 + 8 = -472
     $$
     - Similarly, $$ y = -472 $$ is **not visible** within the y-window.

#### **Graph Description within $$[-10, 10] \times [-10, 10]$$:**

- **Visible Portion:**
  - The vertex at $$ (0.2, 7.4) $$ lies within the window.
  - The y-intercept at $$ (0, 8) $$ is visible.
  - The x-intercepts at approximately $$ (-1.48, 0) $$ and $$ (1.08, 0) $$ are within the window.
  - The parabola will rise to the vertex and then descend, crossing the x-axis at the intercepts.

- **Not Visible Portion:**
  - As $$ x $$ approaches $$ \pm10 $$, the function values drop far below the y-window limit of -10, so these parts of the parabola are **not visible**.

---

### **(b) Completeness of the Graph Within the Given Window**

**Understanding "Complete Graph":**

Assuming "complete graph" refers to the graph of the function being fully depicted within the specified window, we can analyze the completeness based on the window's ability to capture all significant aspects of the graph.

#### **Analysis:**

1. **Vertical Extent ($$ y $$-values):**
   - The function has a **maximum** value at the vertex $$ y = 7.4 $$.
   - As $$ |x| $$ increases beyond the vertex, $$ y $$ decreases without bound (approaches negative infinity).
   - However, the y-window is restricted to $$[-10, 10]$$, so any $$ y $$-values below -10 are **not displayed**.

2. **Horizontal Extent ($$ x $$-values):**
   - The window spans from $$ x = -10 $$ to $$ x = 10 $$.
   - The significant features (vertex, intercepts) lie well within this range.
   
3. **Overall Completeness:**
   - **Visible Features:**
     - Vertex at $$ (0.2, 7.4) $$
     - Y-intercept at $$ (0, 8) $$
     - X-intercepts at approximately $$ (-1.48, 0) $$ and $$ (1.08, 0) $$
     
   - **Missing Features:**
     - The tails of the parabola extend beyond $$ y = -10 $$, which are not visible in the window.
     - Therefore, while all critical points within the window are displayed, the graph is **not fully complete** because portions of the parabola are cut off.

#### **Conclusion:**

**No, the window $$[-10, 10] \times [-10, 10]$$ does **not** provide a complete graph of the function $$ g(x) = -5x^{2} - 2x + 8 $$**. While it captures all the essential features like the vertex and intercepts, the ends of the parabola extend beyond the y-limits of the window and thus are not displayed.

---

If you need a visual representation, consider using graphing tools like [Desmos](https://www.desmos.com/calculator) or [GeoGebra](https://www.geogebra.org/graphing) to plot the function within the specified window.
The graph of $$ g(x) = -5x^{2} - 2x + 8 $$ within the window $$[-10, 10] \times [-10, 10]$$ shows the vertex at $$ (0.2, 7.4) $$, y-intercept at $$ (0, 8) $$, and x-intercepts at approximately $$ (-1.48, 0) $$ and $$ (1.08, 0) $$. However, the graph is **not complete** because parts of the parabola extend beyond the y-window and are not visible.

\( 1<\quad \) (a) Graph the following quadratic function on \( [-10,10] \) by \( [-10,10] \) \( g(x)=-5 x^{2}-2 x+8 \) (b) Does this window give a corplete graph?

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