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A quadratic function is a type of polynomial function where the highest exponent of the variable is 2. These functions are typically written in the form \[ f(x) = ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola. If \( a \) is positive, the parabola opens upward, and if \( a \) is negative, it opens downward. In the given exercise, the function \( f(x) = 1 - x^2 \) is a quadratic function with \( a = -1 \), signifying an upside-down parabola. Understanding the basic shape of quadratic functions helps in graphing and analyzing their properties.

The horizontal-line test is a method used to determine if a function is one-to-one. A function is considered one-to-one if every horizontal line intersects the graph at most once. This ensures that no two different input values (x-values) produce the same output (y-value).

Here's how you can perform the test:

• Draw several horizontal lines across your graph.
• Observe where these lines intersect the graph.

If any horizontal line cuts through the graph more than once, the function is not one-to-one. In our specific case, the function \( f(x) = 1 - x^2 \), horizontal lines intersect the parabola at two points unless they touch the vertex. Therefore, the function fails the horizontal-line test and is not one-to-one. Understanding this test is crucial for identifying whether functions have unique mappings.

The vertex of a parabola is its highest or lowest point, which depends on the direction the parabola opens. It is a critical point in understanding the graph of a quadratic function. For the quadratic equation \( ax^2 + bx + c \), the vertex \((h, k)\) can be found using:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

In the given function \( f(x) = 1 - x^2 \), b = 0, meaning the vertex is at: \( h = 0 \). Substituting \( x = 0 \) into the function gives \( k = 1 \), so the vertex is \( (0, 1) \). The vertex provides important information for graphing the parabola, serving as a reference point for symmetry and height. Knowing the vertex facilitates accurate plotting and helps indicate the general behavior of the function.

Graphing functions is a key skill in understanding their behavior and properties. For a quadratic function like \( f(x) = 1 - x^2 \), graphing involves plotting and connecting points around the vertex. Here’s a simple guide:

• Identify the vertex. For \( f(x) = 1 - x^2 \), the vertex is \( (0, 1) \).
• Plot points around the vertex. For example, at \( x = 1 \) and \( x = -1 \), \( f(1) = f(-1) = 0 \).
• Continue to plot additional points like \( x = 2 \) and \( x = -2 \), which yield \( f(2) = f(-2) = -3 \).

When these points are plotted, connect them smoothly, ensuring the curve reflects the shape of a downward parabola. This method allows students to visualize and deeply understand the nature of quadratic functions.