Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Transformations

College Algebra

3. Functions

Transformations

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1:57 minutes

Problem 84

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x+3|

Verified step by step guidance

Start by graphing the basic absolute value function, \( f(x) = |x| \). This graph is a V-shaped graph with its vertex at the origin (0,0) and opens upwards.

Identify the transformation needed to graph \( g(x) = |x+3| \). Notice that \( |x+3| \) is a horizontal shift of the basic absolute value function.

Understand that the expression \( |x+3| \) means you are shifting the graph of \( |x| \) to the left by 3 units. This is because adding 3 inside the absolute value function moves the graph in the opposite direction.

Apply the transformation: Take each point on the graph of \( f(x) = |x| \) and move it 3 units to the left to obtain the graph of \( g(x) = |x+3| \).

Plot the new graph. The vertex of the graph \( g(x) = |x+3| \) will now be at (-3,0), and the graph will maintain its V-shape, opening upwards.

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This means that for any input x, the function returns x if x is positive or zero, and -x if x is negative. The graph of this function is a V-shape, with its vertex at the origin (0,0), and it is symmetric about the y-axis.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function g(x) = |x + 3|, the graph of f(x) = |x| is shifted horizontally to the left by 3 units. Understanding these transformations is crucial for accurately graphing functions derived from basic forms.

Horizontal Shifts

A horizontal shift occurs when a function is modified by adding or subtracting a constant to the input variable. In the case of g(x) = |x + 3|, the '+3' indicates a shift to the left by 3 units. This concept is essential for predicting how the graph of a function will change based on alterations to its equation.