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

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.