Math Problem Statement

Match the given function to its graph. ​1) y equals sine left parenthesis x minus StartFraction pi Over 2 EndFraction right parenthesis

​ 2) y equals cosine left parenthesis x plus StartFraction pi Over 2 EndFraction right parenthesis

​3) y equals sine left parenthesis x plus StartFraction pi Over 2 EndFraction right parenthesis ​4) y equals cosine left parenthesis x minus StartFraction pi Over 2 EndFraction right parenthesis A -6 6 -2 2 x y

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 2 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. An oscillating periodic curve goes the full length of the graph. In one period, from left to right, the curve starts at minimum at (negative 4.7, negative 1), and rises to cross the x-axis at negative 3.1 and then reaches a maximum at (negative 1.6, 1), and falls to cross the x-axis at 0.0 and reach a minimum at (1.6, negative 1). All coordinates are approximate. B -6 6 -2 2 x y

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 2 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. An oscillating periodic curve goes the full length of the graph. In one period, from left to right, the curve starts at minimum at (negative 3.1, negative 1), and rises to cross the x-axis at negative 1.6 and then reaches a maximum at (0.0, 1), and falls to cross the x-axis at 1.6 and reach a minimum at (3.1, negative 1). All coordinates are approximate. C -6 6 -2 2 x y

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 2 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. An oscillating periodic curve goes the full length of the graph. In one period, from left to right, the curve starts at a maximum at (negative 3.1, 1), and falls to cross the x-axis at negative 1.6 and then reaches a minimum at (0.0, negative 1), and rises to cross the x-axis at 1.6 and reach a maximum at (3.1, 1). All coordinates are approximate. D -6 6 -2 2 x y

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 2 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. An oscillating periodic curve goes the full length of the graph. In one period, from left to right, the curve starts at a maximum at (negative 4.7, 1), and falls to cross the x-axis at negative 3.1 and then reaches a minimum at (negative 1.6, negative 1), and rises to cross the x-axis at 0.0 and reach a maximum at (1.6, 1). All coordinates are approximate. Question content area bottom Part 1 A. ​1B, 2D,​ 3C, 4A B. ​1A, 2D,​ 3C, 4B C. ​1A, 2B,​ 3C, 4D D. ​1C, 2A,​ 3B, 4D

Solution

Let's match each function with its graph based on phase shifts and key points.

Step-by-Step Analysis

  1. y=sin⁡(x−π2)y = \sin(x - \frac{\pi}{2}):

    • This sine function is shifted right by π2\frac{\pi}{2}, meaning it will behave like a cosine function starting at the origin and crossing the x-axis halfway through each period.
    • This graph likely corresponds to Graph B, where the curve starts at a minimum.

  2. y=cos⁡(x+π2)y = \cos(x + \frac{\pi}{2}):

    • This cosine function is shifted left by π2\frac{\pi}{2}, which makes it behave like a negative sine function.
    • The graph will start at a maximum and then descend. This pattern matches Graph D.

  3. y=sin⁡(x+π2)y = \sin(x + \frac{\pi}{2}):

    • This sine function is shifted left by π2\frac{\pi}{2}, making it resemble a cosine curve.
    • This curve will start at a maximum, which corresponds to Graph C.

  4. y=cos⁡(x−π2)y = \cos(x - \frac{\pi}{2}):

    • This cosine function is shifted right by π2\frac{\pi}{2}, making it resemble a positive sine function.
    • This function will start at a minimum, matching Graph A.

Answer

The correct answer is:

  • 1B, 2D, 3C, 4A

Correct Option:

A. 1B,2D,3C,4A1B, 2D, 3C, 4A

Would you like more details or have any questions?

  1. What is the effect of phase shifts on sine and cosine graphs?
  2. How do amplitude and period affect the graph of a trigonometric function?
  3. What does a horizontal shift do to the graph of a function?
  4. How would you match functions with vertical shifts?
  5. How does changing the sign in front of a trigonometric function affect its graph?

Tip: Visualizing phase shifts as horizontal translations helps when identifying function transformations.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine and Cosine Functions
Phase Shifts
Graph Transformations

Formulas

y = sin(x ± a)
y = cos(x ± a)

Theorems

Phase Shift in Trigonometric Functions
Graphing Transformations of Trigonometric Functions

Suitable Grade Level

Grades 9-12

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