f) Sketch the graph. Use the graphing tool to graph the equation $$ y=-x^{2}+1 $$ $$ \begin{array}{c} ext { Click to } \ ext { enlarge } \ ext { graph }\end{array} $$

Question

f) Sketch the graph. Use the graphing tool to graph the equation $$ y=-x^{2}+1 $$ $$ \begin{array}{c}	ext { Click to } \ 	ext { enlarge } \ 	ext { graph }\end{array} $$

UpStudy AI · Accepted Answer

It seems that I am unable to directly plot the graph at this moment. However, I can guide you on how to sketch the graph of the equation $$ y = -x^2 + 1 $$.

### Steps to Sketch the Graph:

1. **Identify the Type of Function**: 
   - The equation $$ y = -x^2 + 1 $$ is a quadratic function, which typically forms a parabola.

2. **Determine the Vertex**:
   - The vertex of the parabola can be found using the formula $$ (h, k) $$ where $$ h = -\frac{b}{2a} $$ for the standard form $$ ax^2 + bx + c $$.
   - Here, $$ a = -1 $$, $$ b = 0 $$, and $$ c = 1 $$.
   - Thus, $$ h = -\frac{0}{2 \cdot -1} = 0 $$.
   - The $$ k $$ value is $$ f(0) = -0^2 + 1 = 1 $$.
   - Therefore, the vertex is at $$ (0, 1) $$.

3. **Determine the Axis of Symmetry**:
   - The axis of symmetry is the vertical line that passes through the vertex, which is $$ x = 0 $$.

4. **Find the Y-Intercept**:
   - The y-intercept occurs when $$ x = 0 $$. From the equation, $$ y = -0^2 + 1 = 1 $$. So, the y-intercept is $$ (0, 1) $$.

5. **Find the X-Intercepts**:
   - To find the x-intercepts, set $$ y = 0 $$:
     $$
     0 = -x^2 + 1 \implies x^2 = 1 \implies x = \pm 1
     $$
   - The x-intercepts are $$ (1, 0) $$ and $$ (-1, 0) $$.

6. **Sketch the Graph**:
   - Plot the vertex $$ (0, 1) $$, the y-intercept $$ (0, 1) $$, and the x-intercepts $$ (1, 0) $$ and $$ (-1, 0) $$.
   - Since the parabola opens downwards (as indicated by the negative coefficient of $$ x^2 $$), draw a symmetric curve through these points.

### Summary of Key Points:
- Vertex: $$ (0, 1) $$
- Y-Intercept: $$ (0, 1) $$
- X-Intercepts: $$ (1, 0) $$ and $$ (-1, 0) $$
- Axis of Symmetry: $$ x = 0 $$

You can use graphing software or a graphing calculator to visualize this function more accurately.
To sketch the graph of $$ y = -x^2 + 1 $$: 1. **Vertex**: Plot at $$ (0, 1) $$. 2. **Y-Intercept**: Also at $$ (0, 1) $$. 3. **X-Intercepts**: At $$ (1, 0) $$ and $$ (-1, 0) $$. 4. **Axis of Symmetry**: The vertical line $$ x = 0 $$. 5. **Shape**: A downward-opening parabola symmetric about $$ x = 0 $$. Plot these points and draw the parabola accordingly.

f) Sketch the graph. Use the graphing tool to graph the equation \( y=-x^{2}+1 \) \( \begin{array}{c}\text { Click to } \\ \text { enlarge } \\ \text { graph }\end{array} \)

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