Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=3 x^2+1

Name: Problem 19, Chapter 11: Beginning and Intermediate Algebra
Uploaded: 2022-02-15T22:45:55-08:00
Duration: 1 min 18 s
Channel: Jessica Kluck
Description: Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=3 x^2+1

step 1 I want to take a look at the function f of x equals 3x squared plus one. From this, I want to de

Graph each parabola. Give the vertex, axis of symmetry,...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial, commonly expressed in the form f(x)=ax²+bx+c, where a, b, and c are constants and a is not zero. Its graph is a parabola, and this structure is fundamental in algebra and calculus.

Parabola

A parabola is the geometric representation of a quadratic function. It is defined by its U-shaped curve, and its orientation (opening upward or downward) is determined by the coefficient of the x² term. The symmetry and shape of the parabola are central to understanding quadratic relationships.

Vertex of a Parabola

The vertex is the highest or lowest point on a parabola, serving as its turning point. It represents the maximum or minimum value of a quadratic function and is found at (-b/(2a), f(-b/(2a))) when the function is in standard form. Identifying the vertex is crucial for graphing and analyzing the function.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two congruent halves. Its equation is x = -b/(2a) for a quadratic function in standard form, and it is important for understanding the balanced nature of the parabola.

Domain and Range

The domain of a function encompasses all the permissible x-values, while the range includes all possible y-values. For quadratic functions, the domain is typically all real numbers. The range is determined by the vertex and the direction the parabola opens (upward or downward), indicating either a minimum or maximum value.

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