Textbook Question
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
293
views
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.2.29
Shifting Graphs
Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.
y = x³ Left 1, down 1
Verified step by step guidance1
Identify the original function: The given function is \( y = x^3 \).
Determine the horizontal shift: A shift to the left by 1 unit means replacing \( x \) with \( x + 1 \) in the function. This gives us \( y = (x + 1)^3 \).
Determine the vertical shift: A shift down by 1 unit means subtracting 1 from the entire function. This modifies the equation to \( y = (x + 1)^3 - 1 \).
Write the equation for the shifted graph: The new equation after applying both shifts is \( y = (x + 1)^3 - 1 \).
Sketch the graphs: Draw the original graph of \( y = x^3 \) and the shifted graph \( y = (x + 1)^3 - 1 \) on the same set of axes, labeling each graph with its respective equation.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graph Shifting
Graph shifting involves translating a graph horizontally and vertically on the coordinate plane. A horizontal shift moves the graph left or right, while a vertical shift moves it up or down. For the equation y = x³, shifting left by 1 unit and down by 1 unit results in the new equation y = (x + 1)³ - 1.
Recommended video:
5:53
Graph of Sine and Cosine Function
Horizontal Shift
A horizontal shift changes the x-values of a function. If a graph is shifted left by 'a' units, the transformation is represented by replacing x with (x + a) in the equation. For y = x³, shifting left by 1 unit modifies the equation to y = (x + 1)³, effectively moving the graph one unit to the left.
Recommended video:
5:25Intro to Transformations
Vertical Shift
A vertical shift affects the y-values of a function. Shifting a graph down by 'b' units involves subtracting 'b' from the entire function. In the case of y = x³, shifting down by 1 unit results in the equation y = x³ - 1, which lowers the graph by one unit on the y-axis.
Recommended video:
5:25Intro to Transformations
Related Practice
Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
246
views
Textbook Question
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
356
views
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
240
views
Textbook Question
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
534
views
Textbook Question
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = ½ sin (πx – x) + ½
134
views