Let's break down the function $y = 12 \cos\left( \frac{\pi x}{4} \right)$ to determine the key points over one period.

1. Amplitude and Period

• The amplitude is the coefficient of the cosine function, which is $12$. So, the amplitude is $12$.
• The period is calculated using the formula:

$\text{Period} = \frac{2\pi}{B}$ Where $B$ is the coefficient of $x$ inside the cosine. In this case, $B = \frac{\pi}{4}$. Thus:

$\text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8$

2. Key Points of One Period

Cosine functions have 5 key points over one period: starting at a maximum, passing through intercepts (where $y = 0$), reaching a minimum, and returning to the maximum. For $\cos\left(\frac{\pi x}{4}\right)$, let's calculate the key points.

• The general cosine function starts at $x = 0$ (maximum), reaches zero at $\frac{1}{4}$ of the period, minimum at $\frac{1}{2}$ of the period, zero again at $\frac{3}{4}$ of the period, and maximum at the end of the period.

Given that the period is 8, the key points occur at:

• Start (Maximum): At $x = 0$, $y = 12$. $(x, y) = (0, 12)$
• First Intercept: At $x = 2$, $y = 0$. $(x, y) = (2, 0)$
• Minimum: At $x = 4$, $y = -12$. $(x, y) = (4, -12)$
• Second Intercept: At $x = 6$, $y = 0$. $(x, y) = (6, 0)$
• End (Maximum): At $x = 8$, $y = 12$. $(x, y) = (8, 12)$

Summary of Key Points

• Maximum: $(0, 12)$
• First intercept: $(2, 0)$
• Minimum: $(4, -12)$
• Second intercept: $(6, 0)$
• Maximum again: $(8, 12)$

Incorrect Answers

Your previously marked incorrect points were likely due to incorrect $x$-values or $y$-values at these key points. With these corrected key points, you should have the proper intercepts and extrema.

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Would you like further details on this solution, or do you have any questions?

Related Questions:

1. How do we determine the phase shift in cosine functions?
2. Can you explain how transformations like vertical shifts affect key points?
3. What is the significance of the period in oscillating functions?
4. How would the key points change if the function was $y = -12 \cos\left(\frac{\pi x}{4}\right)$?
5. How do we graph the function using the key points?

Tip: In trigonometric functions, always confirm the period and amplitude before plotting or calculating key points!