What is the maximum y value obtained for the graph of
𝑦
=
3
cos
⁡
(
2
𝑥
+
1
)

Question

What is the maximum y value obtained for the graph of 
𝑦
=
3
cos
⁡
(
2
𝑥
+
1
)

BlackTom AI · Accepted Answer

Cosine function has an amplitude equal to the absolute value of its leading coefficient, so the maximum and minimum values are determined by that amplitude multiplied by the range of cosine, which is [-1, 1].

1. Relevant knowledge points and concepts
- The basic cosine function cos(t) takes values between -1 and 1 for all real t.
- If y = A cos(Bx + C), the amplitude is |A|, so the range of y is [−|A|, |A|].
- Horizontal shifts and stretches (parameters B and C) do not change the amplitude; they only change where maxima/minima occur on the x-axis.

2. Step-by-step reasoning
- Given y = 3 cos(2x + 1). Here A = 3, B = 2, C = 1.
- Since cos(θ) ∈ [−1, 1] for all θ, multiply by 3 to get y ∈ [−3, 3].
- Therefore the maximum possible y-value is 3.

3. Intermediate calculation steps (showing existence of the maximum)
- Maximum occurs when cos(2x + 1) = 1.
- Solve 2x + 1 = 2πk, for integers k:
  2x = 2πk − 1
  x = πk − 1/2
- For k = 0, x = −1/2 gives cos(2(−1/2) + 1) = cos(0) = 1 and y = 3(1) = 3, so the maximum 3 is attained.

4. Why the correct answer is correct
- The coefficient 3 sets the amplitude, so the function's outputs are confined between −3 and 3; the highest achievable value is therefore 3. Horizontal phase shift (the +1 inside) or frequency change (the factor 2) cannot increase this amplitude.

5. Why other answers would be incorrect
- Any value greater than 3 is impossible because cos cannot exceed 1 and multiplying by 3 caps the value at 3. Any value less than 3 is not the maximum (though it may be attained at other x), so it would be incorrect as the maximum.

Conclusion: The maximum y-value is 3, which matches the provided answer.

What is the maximum y value obtained for the graph of 𝑦 = 3 cos ⁡ ( 2 𝑥 + 1 )单项选择题

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