Textbook Question
Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. (2²)⁵
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 3
Fill in the blank(s) to correctly complete each sentence.
The graph of ƒ(x) = (x + 4)² is obtained by shifting the graph of y = x² to the ___ 4 units.
Verified step by step guidance1
Identify the base function: The given function is \( f(x) = (x + 4)^2 \), which is a transformation of the parent function \( y = x^2 \).
Recall the effect of horizontal shifts on the graph of \( y = x^2 \): Replacing \( x \) with \( x + h \) shifts the graph horizontally by \( h \) units.
Determine the direction of the shift: Since the function is \( (x + 4)^2 \), the graph shifts horizontally to the left by 4 units (because adding inside the parentheses moves the graph left).
Write the completed sentence: The graph of \( f(x) = (x + 4)^2 \) is obtained by shifting the graph of \( y = x^2 \) to the left 4 units.
Understand that horizontal shifts do not affect the shape of the parabola, only its position along the x-axis.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformations
Function transformations involve shifting, stretching, or reflecting the graph of a base function. In this case, adding a constant inside the function's argument shifts the graph horizontally. Understanding how these changes affect the graph is essential for interpreting and sketching functions.
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4:22
Domain and Range of Function Transformations
Horizontal Shifts in Quadratic Functions
A horizontal shift occurs when a constant is added or subtracted inside the function's input, such as (x + h)². Specifically, f(x) = (x + 4)² shifts the graph of y = x² horizontally by 4 units. The sign inside the parentheses determines the direction of the shift.
Recommended video:
6:31Phase Shifts
Graph of the Basic Quadratic Function y = x²
The graph of y = x² is a parabola centered at the origin with its vertex at (0,0). It opens upwards and is symmetric about the y-axis. Recognizing this base graph helps in understanding how transformations like shifts affect its position.
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6:22Graphs of Secant and Cosecant Functions
Related Practice
Textbook Question
CONCEPT PREVIEW Which of the following is the correct factorization of x⁴ - 1? A. (x² - 1) (x² + 1) B. (x² + 1) (x + 1) (x - 1) C. (x² - 1)² D. (x - 1)² (x + 1)²
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Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The opposite, or negative, of a number is its _______.
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Textbook Question
Rewrite each expression using the distributive property and simplify, if possible. See Example 7. x + x
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Textbook Question
Simplify each expression. See Example 8. 10x (3)(y)
Textbook Question
Find the given distances between points P, Q, R, and S on a number line, with coordinates -4, -1, 8, and 12, respectively. See Example 3. d (Q, R)
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