Analyzing Graph Transformations and Intersection Points

00:00 10 marks

Consider the function \(f(x) = 2|x – 3| + 1\). You are required to perform a series of transformations and solve for the intersection with the line \(y = 5\) analytically and using a graphing calculator (GDC).

Part a

4 minutes 3 marks

Question

Sketch the transformations that occur when the parent function \(y = |x|\) is transformed into \(f(x) = 2|x – 3| + 1\), detailing the effect of each transformation.

Hint Hint

Hint

Identify the horizontal shift, vertical stretch, and vertical shift, and describe the order in which they are applied to \(y=|x|\).

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Complete Answer

The transformation involves a horizontal shift to the right by 3 units, a vertical stretch by a factor of 2, and a vertical shift upward by 1 unit.

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Part b

3 minutes 3 marks

Question

Using a graphing calculator (GDC), determine the x-coordinate(s) where \(f(x) = 2|x – 3| + 1\) intersects the line \(y = 5\). State the method used on the GDC.

Hint Hint

Hint

Input \(y = 2|x – 3| + 1\) and \(y = 5\) into the GDC. Use the ‘intersect’ function to find the intersection points.

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Complete Answer

The function \(f(x)\) intersects \(y = 5\) at \(x = 1\) and \(x = 5\).

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Part c

3 minutes 4 marks

Question

For HL: Explain why the GDC solution is reasonable by comparing it with the analytical solution. Solve \(f(x) = 5\) analytically to support your reasoning.

Hint Hint

Hint

Solve \(2|x-3|+1 = 5\) algebraically. Consider the symmetry of the absolute value function around \(x=3\).

Answer Answer
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Complete Answer

The GDC solution is reasonable because it matches the analytical solution. Analytically solving \(2|x-3|+1 = 5\) gives \(x = 1\) and \(x = 5\), which are symmetric around the vertex of \(f(x)\) at \(x = 3\).

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