Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 55

Chapter 3, Problem 55

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

Verified step by step guidance

Start by understanding the parent function, which is the standard quadratic function \(f(x) = x^{2}\). This graph is a parabola with its vertex at the origin \((0,0)\) and it opens upwards.

Identify the given function \(g(x) = (x - 2)^{2}\). Notice that it is in the form \(g(x) = (x - h)^{2}\), where \(h = 2\).

Recognize that the transformation from \(f(x)\) to \(g(x)\) involves a horizontal shift. Specifically, the graph of \(f(x)\) is shifted to the right by 2 units because of the \((x - 2)\) inside the square.

To graph \(g(x)\), take the graph of \(f(x) = x^{2}\) and move every point 2 units to the right. The vertex of \(g(x)\) will now be at \((2, 0)\) instead of \((0, 0)\).

Check a few points to confirm the transformation: for example, when \(x = 2\), \(g(2) = (2 - 2)^{2} = 0\), which matches the new vertex. Plot these 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.

Standard Quadratic Function

The standard quadratic function is f(x) = x², which produces a parabola opening upwards with its vertex at the origin (0,0). Understanding this basic graph is essential as it serves as the starting point for applying transformations.

Horizontal Shifts of Functions

A horizontal shift moves the graph left or right along the x-axis. For g(x) = (x − 2)², the graph of f(x) = x² shifts 2 units to the right, changing the vertex from (0,0) to (2,0). This shift does not affect the shape of the parabola.

Graph Transformations

Graph transformations involve changing a function's graph through shifts, stretches, compressions, or reflections. Recognizing how each transformation affects the graph helps in sketching the new function accurately from the original.

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