Part 1The function graphed is of the form y equals a sine bxy=asinbx or y equals a cosine bxy=acosbx, where b 0. Determine the equation of the graph. Part 1
-44xy00piπ2 pi2π Edit coordinates (0,0) A coordinate system has a horizontal x-axis labeled from 0 to 2 pi in increments of StartFraction pi Over 2 EndFraction and a vertical y-axis labeled from negative 4 to 4 in increments of 1. A curve rises from a minimum at (0, negative 2) to a point at (2 pi, 0). Part 1yequals=[input]enter your response here (Use integers or fractions for any numbers in the expression.)

Question

Part 1The function graphed is of the form y equals a sine bxy=asinbx or y equals a cosine bxy=acosbx​, where b​ > 0. Determine the equation of the graph. Part 1
-44xy00piπ2 pi2π Edit coordinates (0,0) A coordinate system has a horizontal x-axis labeled from 0 to 2 pi in increments of StartFraction pi Over 2 EndFraction and a vertical y-axis labeled from negative 4 to 4 in increments of 1. A curve rises from a  minimum at (0, negative 2) to a point at (2 pi, 0). Part 1yequals=[input]enter your response here ​(Use integers or fractions for any numbers in the​ expression.)

BlackTom AI · Accepted Answer

The prompt provides a graph-based question asking for a function of the form y = a sin(bx) or y = a cos(bx) with b > 0, given that the curve has a minimum at (0, -2) and passes through (2π, 0). Since there are no multiple-choice options provided in this question, I will analyze the given answer and show how to determine the correct equation step by step, and explain why the stated answer (if it were y = -2 cos(x)) would not fit the graph as described.

First, evaluate what the graph tells us about the sinusoid:
- A minimum at x = 0 with y = -2 suggests the function at x = 0 equals the minimum value, i.e., the amplitude times the cosine/sine value at zero yields -2. For a cosine form y = a cos(bx), cos(0) = 1, so y(0) = a; thus a must be -2 to produce the minimum -2 at x = 0. For a sine form y = a sin(bx), sin(0) = 0, which would give y(0) = 0, not -2, so the sine form is inconsistent with a minimum at x = 0 unless there is a vertical shift, which is not indicated here. Therefore, the cosine form is the appropriate starting point: y = a cos(bx) with a = -2.

- Now use the second provided point (2π, 0). If y = -2 cos(bx), then y(2π) = -2 cos(2π b) should equal 0. This means cos(2π b) must be 0, i.e., 2π b = π/2 + kπ for some integer k. The simplest positive choice is 2π b = π/2, giving b = 1/4. With b = 1/4, the equation is y = -2 cos(x/4). Then at x = 2π, cos(2π * (1/4)) = cos(π/2) = 0, so y = 0, matching the graph point.

- Therefore, the combination a = -2 and b = 1/4 yields the expected graph behavior: a minimum at x = 0 of -2 and a value of 0 at x = 2π. In general, the period of y = a cos(bx) is 2π/b, which here would be 2π/(1/4) = 8π, and the given information only specified one point and the minimum; the derived period is consistent with a standard cosine wave scaled horizontally to fit the given data.

- Contrast with the proposed answer y = -2 cos(x): If we substitute x = 2π into this expression, we get y = -2 cos(2π) = -2, not 0. That clearly contradicts the stated graph point (2π, 0). Hence, y = -2 cos(x) does not fit the provided data and is incorrect for this graph.

- If there were any other hidden constraints (for example, a vertical shift or a different phase), we would adjust accordingly. However, given the specified minimum at x = 0 and the point (2π, 0), the simplest exact fit without vertical shift is y = -2 cos(x/4).

In summary, based on the described graph, the correct sinusoid form is y = -2 cos(x/4). The candidate y = -2 cos(x) fails to satisfy the point (2π, 0); hence it is not consistent with the graph.

Homework:Section 6.3 Homework

查看解析

登录即可查看完整答案

类似问题

Which equation best describes this function?

An oscillating voltage compressed horizontally is modeled by:Determine the domain, range, and period of the function if 0 < x < 2π

Given the function: What are the amplitude, period, and phase shift?

Which statement best describes the period of ?

If and is in Quadrant II, what is ?

Using the relationship between the sine function and its reciprocal, which of the following describes the restrictions on the domain of when 0 < x < 2π

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单