Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 19

Chapter 3, Problem 19

use the graph of y = f(x) to graph each function g.

g(x) = f(x-1)

Verified step by step guidance

Identify the given function and the transformation: The original function is \(y = f(x)\), and the new function is \(g(x) = f(x - 1)\). This represents a horizontal shift of the graph of \(f(x)\).

Understand the effect of \(f(x - 1)\): Replacing \(x\) with \(x - 1\) shifts the graph of \(f(x)\) to the right by 1 unit. This means every point on the graph of \(f(x)\) moves 1 unit to the right to get the graph of \(g(x)\).

Apply the horizontal shift to key points: Take the key points from the original graph of \(f(x)\), which are \((-4, 0)\), \((0, -16)\), and \((4, 0)\). Add 1 to each x-coordinate to find the corresponding points on \(g(x)\).

Calculate the new points for \(g(x)\): The points become \((-4 + 1, 0) = (-3, 0)\), \((0 + 1, -16) = (1, -16)\), and \((4 + 1, 0) = (5, 0)\).

Sketch the graph of \(g(x)\): Plot the new points \((-3, 0)\), \((1, -16)\), and \((5, 0)\) on the coordinate plane and draw a smooth parabola through these points, maintaining the shape of the original graph but shifted right by 1 unit.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Function Transformation - Horizontal Shift

A horizontal shift of a function occurs when the input variable x is replaced by (x - h), shifting the graph h units to the right if h > 0, or to the left if h < 0. For g(x) = f(x - 1), the graph of f(x) moves 1 unit to the right.

Graphing Quadratic Functions

Quadratic functions form parabolas, typically expressed as y = ax^2 + bx + c. Key points such as vertex and x-intercepts help in sketching the graph. The given graph shows a parabola with vertex at (0, -16) and x-intercepts at (-4, 0) and (4, 0).

Interpreting Key Points on a Graph

Identifying and using key points like intercepts and vertex is essential for graph transformations. These points help visualize how the graph shifts or changes shape. For g(x) = f(x - 1), each key point of f(x) moves 1 unit right, e.g., vertex moves from (0, -16) to (1, -16).

Recommended video:

Guided course

04:29

Graphing Equations of Two Variables by Plotting Points

Related Practice

Textbook Question

In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²

Textbook Question

Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph.

Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-1/4, -1/7) and (3/4, 6/7)

views

Textbook Question

Find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)

views

Textbook Question

The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x+2)³

Textbook Question

Find the domain of each function. f(x) = 1/√(x - 3)

views

use the graph of y = f(x) to graph each function g.g(x) = f(x-1)

Verified video answer for a similar problem:

Key Concepts

Function Transformation - Horizontal Shift

Graphing Quadratic Functions

Interpreting Key Points on a Graph

use the graph of y = f(x) to graph each function g.

g(x) = f(x-1)