d) \( f(x)=-\frac{1}{4}(x+1)^{2}+5 \)

To analyze the function \( f(x) = -\frac{1}{4}(x+1)^{2} + 5 \), we can identify several features of the function, such as its vertex, axis of symmetry, and whether it opens upwards or downwards. 1. **Identify the vertex**: The function is in vertex form \( f(x) = a(x-h)^{2} + k \), where \( (h, k) \) is the vertex. Here, \( a = -\frac{1}{4} \), \( h = -1 \), and \( k = 5 \). Thus, the vertex is at the point \( (-1, 5) \). 2. **Determine the direction of opening**: Since \( a < 0 \) (specifically, \( a = -\frac{1}{4} \)), the parabola opens downwards. 3. **Find the axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex, given by \( x = h \). Therefore, the axis of symmetry is \( x = -1 \). 4. **Calculate the y-intercept**: To find the y-intercept, we set \( x = 0 \): \[ f(0) = -\frac{1}{4}(0+1)^{2} + 5 = -\frac{1}{4}(1) + 5 = -\frac{1}{4} + 5 = \frac{19}{4} \] Thus, the y-intercept is \( \left(0, \frac{19}{4}\right) \). 5. **Find the x-intercepts**: To find the x-intercepts, we set \( f(x) = 0 \): \[ 0 = -\frac{1}{4}(x+1)^{2} + 5 \] Rearranging gives: \[ \frac{1}{4}(x+1)^{2} = 5 \] Multiplying both sides by 4: \[ (x+1)^{2} = 20 \] Taking the square root of both sides: \[ x + 1 = \pm \sqrt{20} \] Thus: \[ x = -1 \pm 2\sqrt{5} \] Therefore, the x-intercepts are \( x = -1 + 2\sqrt{5} \) and \( x = -1 - 2\sqrt{5} \). Now, let's summarize the findings: - Vertex: \( (-1, 5) \) - Opens: Downwards - Axis of symmetry: \( x = -1 \) - Y-intercept: \( \left(0, \frac{19}{4}\right) \) - X-intercepts: \( x = -1 + 2\sqrt{5} \) and \( x = -1 - 2\sqrt{5} \) If you need further analysis or specific calculations, please let me know!