Consider the given function \( g(x)=-(x-2)^{2}+4 \). (a) Determine whether the graph of the parabola opens upward or downward. (b) Identify the vertex. (c) Determine the \( x \)-intercept(s). (d) Determine the \( y \)-intercept(s). (e) Sketch the function. (f) Determine the axis of symmetry. (g) Determine the minimum or maximum value of the function. (h) Write the domain and range in interval notation. Write your answers in exact form.

**(a) Opening Direction** The function is given by \[ g(x) = - (x-2)^2 + 4. \] The quadratic term \((x-2)^2\) is multiplied by \(-1\), so the parabola opens **downward**. --- **(b) Vertex** The function is in vertex form \[ g(x) = a(x-h)^2 + k, \] with \(a = -1\), \(h = 2\), and \(k = 4\). Thus, the vertex is \[ (2, 4). \] --- **(c) \(x\)-Intercept(s)** Set \(g(x) = 0\): \[ - (x-2)^2 + 4 = 0. \] Solve for \((x-2)^2\): \[ -(x-2)^2 = -4 \quad \Longrightarrow \quad (x-2)^2 = 4. \] Taking the square root: \[ x-2 = \pm 2. \] Hence, \[ x = 2 \pm 2. \] So the \(x\)-intercepts are at: \[ x = 0 \quad \text{and} \quad x = 4. \] Thus, the points are: \[ (0,0) \quad \text{and} \quad (4,0). \] --- **(d) \(y\)-Intercept(s)** Set \(x = 0\): \[ g(0) = - (0-2)^2 + 4 = -4 + 4 = 0. \] Thus, the \(y\)-intercept is: \[ (0,0). \] --- **(e) Sketch of the Function** - The parabola opens **downward**. - The **vertex** is at \((2, 4)\). - It has **\(x\)-intercepts** at \((0,0)\) and \((4,0)\) as well as a **\(y\)-intercept** at \((0,0)\). - The **axis of symmetry** is the vertical line through \(x = 2\). A rough sketch would show a downward opening parabola with its highest point at \((2,4)\). The arms of the parabola pass through \((0,0)\) and \((4,0)\). --- **(f) Axis of Symmetry** The axis of symmetry is the vertical line through the vertex: \[ x = 2. \] --- **(g) Maximum or Minimum Value** Since the parabola opens downward, it has a **maximum** value. The vertex \((2, 4)\) gives the maximum value, so the maximum value is: \[ 4. \] --- **(h) Domain and Range** - **Domain:** For any quadratic function, the domain is all real numbers: \[ (-\infty, \infty). \] - **Range:** Since the maximum value is \(4\) and the parabola opens downward, the range is: \[ (-\infty, 4]. \]