To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x^2? What is its domain?

Verified step by step guidance

1. Step 1: The graph of the function ƒ(x)=x^2 is a parabola that opens upwards. It starts from the origin (0,0) and spreads out on both sides along the x-axis. The lowest point of the graph is at the origin.
2. Step 2: To identify the correct graph, look for a U-shaped curve that starts from the origin and opens upwards. This is the graph of ƒ(x)=x^2.
3. Step 3: The domain of a function is the set of all possible x-values that can be plugged into the function. For the function ƒ(x)=x^2, you can plug in any real number for x and get a real number for ƒ(x).
4. Step 4: Therefore, the domain of the function ƒ(x)=x^2 is all real numbers, which can be written in interval notation as (-∞, ∞).
5. Step 5: In conclusion, the graph of ƒ(x)=x^2 is a parabola that opens upwards and its domain is all real numbers.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form ƒ(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which opens upwards if a is positive and downwards if a is negative. For the function ƒ(x) = x², the graph is a parabola that opens upwards with its vertex at the origin (0,0).

Graphing Parabolas

When graphing a quadratic function like ƒ(x) = x², key features include the vertex, axis of symmetry, and direction of opening. The vertex is the lowest point for upward-opening parabolas, and the axis of symmetry is the vertical line that passes through the vertex. The shape of the graph is determined by the coefficient of x², which influences the width and steepness of the parabola.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the quadratic function ƒ(x) = x², the domain is all real numbers, denoted as (-∞, ∞), because you can substitute any real number for x and obtain a valid output. Understanding the domain is crucial for analyzing the behavior of the function and its graph.