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The vertex of a parabola is a special point that indicates the maximum or minimum of the quadratic function, depending on its opening direction. For quadratic equations of the form \( ax^2 + bx + c \), the vertex can be calculated using the formula \( x = \frac{-b}{2a} \). In the exercise provided, the equation \( f(x) = |-x^2 + 1| \) corresponds to \( a = -1 \) and \( b = 0 \), therefore, the x-coordinate of the vertex is \( x = 0 \).
To discover the y-coordinate, substitute \( x = 0 \) back into the equation, resulting in \( f(0) = |-0^2 + 1| = 1 \). Thus, the vertex is at the point \((0, 1)\). This vertex being above the x-axis signifies the topping point of the "W" shape curve created by the absolute value function applied to the quadratic expression.

X-intercepts are points where the graph of the function crosses the x-axis. In mathematical terms, these are the solutions of the equation \( f(x) = 0 \). For the equation given, \( f(x) = |-x^2 + 1| \), set the inside function to zero, \(-x^2 + 1 = 0 \).
After solving, you find that the x-intercepts occur at \( x = -1 \) and \( x = 1 \), making them \((-1, 0)\) and \((1, 0)\) respectively on the graph. These x-intercepts tell us where the directed graph reflects on the x-axis, creating the mirror-like 'W' pattern.

The y-intercept is the point where the graph crosses the y-axis, which occurs when \( x = 0 \). Finding this is quite simple: substitute \( x = 0 \) in the function \( f(x) = |-x^2 + 1| \).
Conducting this substitution yields \( f(0) = |0^2 + 1| = 1 \). The coordinate \((0, 1)\) represents the y-intercept of the graph. Interestingly, for the given function, this point is both the y-intercept and the vertex. This duality of roles emphasizes its significance in sketching the graph effectively.

An absolute value function is a type of piecewise function that results in non-negative outputs. The absolute value is represented by the symbol \(|\cdot|\). For our function \( f(x) = |-x^2 + 1| \), applying the absolute value transforms the outputs less than zero into their positive counterparts.
The impact of the absolute value on the graph is visually distinctive: parts of the quadratic curve that would dip below the x-axis are flipped or "reflected" upward. This generates a 'W' shaped graph due to the nature of squaring the x-term and applying negation and absolute value.

• The downward shape from \(-x^2\) becomes upward upon applying absolute value.
• Parts of the curve that would normally enter negative y-values remain above the axis.

These characteristics make graphing absolute value functions unique, with all range values being zero or positive.