Functions and Graphs


Find the natural domain and graph the functions in Exercises 15–20.


f(x) = 1 − 2x − x²

Verified step by step guidance

1. Identify the type of function: The given function \( f(x) = 1 - 2x - x^2 \) is a quadratic function. Quadratic functions are polynomials of degree 2.
2. Determine the natural domain: Since \( f(x) \) is a polynomial, it is defined for all real numbers. Therefore, the natural domain of \( f(x) \) is \( (-\infty, \infty) \).
3. Find the vertex of the parabola: The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \). To find the vertex, use the formula \( h = -\frac{b}{2a} \) where \( a = -1 \) and \( b = -2 \).
4. Calculate the vertex: Substitute \( a = -1 \) and \( b = -2 \) into the formula \( h = -\frac{-2}{2(-1)} \) to find the x-coordinate of the vertex. Then, substitute this value back into the function to find the y-coordinate.
5. Sketch the graph: Plot the vertex and a few additional points by choosing x-values and calculating corresponding y-values. Since the coefficient of \( x^2 \) is negative, the parabola opens downwards. Draw the parabola using the vertex and points as a guide.

Key Concepts

Natural Domain

The natural domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like f(x) = 1 - 2x - x², the natural domain is typically all real numbers, as polynomials do not have restrictions such as division by zero or square roots of negative numbers.

Graphing Functions

Graphing a function involves plotting its output values (y-values) against its input values (x-values) on a coordinate plane. For the function f(x) = 1 - 2x - x², this requires determining key points, such as intercepts and turning points, and understanding the shape of the graph, which is a downward-opening parabola due to the negative leading coefficient.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. The function f(x) = 1 - 2x - x² can be rewritten as f(x) = -x² - 2x + 1, revealing its parabolic nature. The vertex, axis of symmetry, and direction of opening are key features that help in graphing and analyzing the function.