Trig functions grade 11 and 12: Determine Equation

Summary

This lesson teaches how to determine the equation of a graph by analyzing its transformations. The instructor explains how graphs can be shifted vertically or horizontally, stretched, or compressed. The key to identifying these transformations is to look for the general form of the graph equation and find a known point to determine the shift. The amplitude, range, domain, and period of the graph are also discussed, with the example of a cosine graph being shifted by 30 degrees to the right, resulting in a final equation of the form y = A*cos(B(x - p)) + C.

Takeaways

• 📈 A graph can be manipulated in four ways: vertical shift, vertical stretch/compress, horizontal shift, and horizontal stretch/compress.
• ⬆️ Vertical shifts can move the graph up or down, and horizontal shifts can move it left or right on the x-axis.
• 🔍 To determine the type of shift, look for the general form of the graph equation and identify any transformations.
• 🟢 When identifying horizontal shifts, focus on the point where the graph intersects the x-axis and compare it to the original graph's intersection point.
• ⚖️ The amplitude of a graph represents its maximum distance from the resting position, indicating the height of the wave's peak.
• 📉 The range of a graph is the extent of its y-values, from the lowest to the highest point on the graph.
• 🔢 The domain of a graph refers to the set of all possible x-values, which can be determined by looking at the graph's given range.
• ⏳ The period of a graph is the length of time it takes for the graph to repeat its pattern, which remains constant even with horizontal shifts.
• 🔑 Understanding the normal appearance of basic graphs is crucial for identifying transformations, such as the original coordinates of a cosine graph.
• 📐 The equation of a transformed graph can be determined by identifying the horizontal shift (p), amplitude, range, and period from the graph.

Q & A

What are the four things a graph can do according to the lesson?

A graph can be shifted vertically, stretched or compressed vertically, shifted horizontally, and stretched or compressed horizontally.

How can you determine if a graph has been shifted vertically?

You can determine a vertical shift by looking at the general form of the graph and identifying any changes in the equation that would result in a vertical movement.

What does a plus sign in the equation of a graph indicate in terms of horizontal shift?

A plus sign in the equation does not necessarily mean the graph is shifted to the left. It could be part of a larger expression that results in a negative value, which would shift the graph to the right.

Why is it important to know the coordinates of an original cosine graph?

Knowing the coordinates of an original cosine graph helps in determining how much a graph has been shifted horizontally by comparing it to a known point on the shifted graph.

What is the significance of the point (300, 0) in determining the horizontal shift of the graph?

The point (300, 0) indicates that the graph has been shifted 30 degrees to the right from its original position, where the cosine graph would normally have a point at (270, 0).

What is the amplitude of the graph described in the lesson?

The amplitude of the graph is one, which is the maximum distance from the resting position.

How do you determine the range of the graph from the information provided?

The range of the graph is determined by identifying the lowest and highest y-values on the graph, which in this case are -1 and 1, respectively.

What is the domain of the graph as described in the lesson?

The domain of the graph is from -180 to 360 degrees.

Why does the period of a cosine graph remain the same even when it is shifted horizontally?

The period of a cosine graph, which is 360 degrees for a normal cosine graph, remains unchanged because shifting horizontally does not affect the time it takes for the graph to repeat.

What is the final equation of the graph after determining the horizontal shift?

The final equation of the graph after determining the horizontal shift is y = cos(x - 30), where p is -30.