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The axis of symmetry is a crucial aspect of understanding any parabola. It is a vertical line that runs through the vertex of the parabola, dividing it into two mirror images. For the given equation \(y = \frac{1}{2}(x-4)^{2}+1\), we can spot the axis of symmetry by looking at the value of \(h\) in the vertex form of a parabola, \(y = a(x-h)^2 + k\).For this equation, \(h = 4\). Therefore, the axis of symmetry is the line \(x = 4\). This line represents the balance point of the parabola where it is symmetrically divided.

The \(x\)-intercepts are the points where the parabola crosses the \(x\)-axis. To find these points, we set \(y\) to \(0\) in the equation and solve for \(x\): \[ 0 = \frac{1}{2}(x-4)^2 + 1 \ Minus 1 from both sides: -1 = \frac{1}{2}(x-4)^2\ Multiply both sides by 2: \ -2 = (x-4)^2 \] The equation \((x-4)^2 = -2\) has no real solutions because a square root of a negative number is not real. Hence, this parabola does not cross the \(x\)-axis and has no \(x\)-intercepts.

To find the \(y\)-intercept of the parabola, we set \(x\) to \(0\) and solve for \(y\). For the equation \(y = \frac{1}{2}(x-4)^2 + 1\): Let \(x = 0\): \[ y = \frac{1}{2}(0-4)^2 + 1 \ y = \frac{1}{2}(16) + 1 \ y = 8 + 1 \ y = 9 \] So, the \(y\)-intercept is \((0, 9)\). This is the point where the parabola crosses the \(y\)-axis.

The focus and directrix are unique components of a parabola. They help to define its shape and orientation. For a parabola in the form \(y = a(x-h)^2 + k\), the focus is located at \((h, k + \frac{1}{4a})\) and the directrix is the line \(y = k - \frac{1}{4a}\).Using the given parameters \(a = \frac{1}{2}\), \(h = 4\), and \(k = 1\), let's calculate these points: \[ \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \ Focus: (4, 1 + \frac{1}{2}) = (4, 1.5) \ Directrix: y = 1 - \frac{1}{2} = 0.5 \]The focus is the point \((4, 1.5)\), and the directrix is the horizontal line \(y = 0.5\). The parabola will always open towards its focus, making this point pivotal in graphing and understanding the parabola's behavior.