Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 53

Chapter 3, Problem 53

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Verified step by step guidance

Start with the standard quadratic function f(x) = x². This is a parabola that opens upwards with its vertex at the origin (0, 0). The graph is symmetric about the y-axis.

Understand the transformation: The given function g(x) = x² - 2 is derived from f(x) = x² by subtracting 2. This represents a vertical shift downward by 2 units.

To graph g(x), take each point on the graph of f(x) = x² and move it 2 units downward. For example, the vertex (0, 0) of f(x) will move to (0, -2) for g(x).

Plot the new points after the vertical shift. For instance, if (1, 1) is a point on f(x), it will become (1, -1) on g(x). Similarly, (-1, 1) will become (-1, -1). Repeat this for other points.

Draw the new parabola for g(x) = x² - 2, ensuring it maintains the same shape as f(x) = x² but is shifted downward by 2 units. Label the vertex and key points to complete the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of quadratic functions is essential for analyzing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For quadratic functions, vertical shifts occur when a constant is added or subtracted from the function, such as in g(x) = x² - 2, which shifts the graph of f(x) = x² down by 2 units. Recognizing these transformations helps in accurately graphing modified functions.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on its orientation. For the standard quadratic function f(x) = x², the vertex is at the origin (0,0). In the function g(x) = x² - 2, the vertex shifts to (0, -2), which is crucial for understanding the graph's position and shape after transformation.

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Horizontal Parabolas

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