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

Common Functions

College Algebra

3. Functions

Common Functions

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1:53 minutes

Problem 107

Textbook Question

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

Verified step by step guidance

Start with the given function \( f(x) = 2x + 5 \).

To translate the graph up 2 units, add 2 to the function: \( f(x) + 2 = 2x + 5 + 2 = 2x + 7 \).

To translate the graph to the left 3 units, replace \( x \) with \( x + 3 \): \( f(x + 3) = 2(x + 3) + 7 \).

Simplify the expression: \( 2(x + 3) + 7 = 2x + 6 + 7 \).

Combine like terms to find \( g(x) = 2x + 13 \).

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Key Concepts

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

Function Translation

Function translation involves shifting the graph of a function without altering its shape. A vertical translation moves the graph up or down, while a horizontal translation shifts it left or right. For example, translating a function f(x) up by k units adds k to the function, resulting in f(x) + k.

Linear Functions

Linear functions are mathematical expressions of the form g(x) = ax + b, where 'a' represents the slope and 'b' the y-intercept. The slope indicates the steepness of the line, while the y-intercept is the point where the line crosses the y-axis. Understanding the characteristics of linear functions is essential for manipulating and translating them.

Combining Translations

When translating a function, multiple transformations can be combined. For instance, to translate f(x) = 2x + 5 up 2 units, you add 2 to the function, resulting in f(x) + 2. To shift it left by 3 units, you replace x with (x + 3). The final function incorporates both transformations, demonstrating how to effectively combine shifts.