Maths Extension 1 Year 11 - Module 1 - Lesson 12
Graphical Relationships: Absolute Value
Name
Date
Class
1. Lesson focus
Graphical relationship y=|f(x)|. The absolute value keeps non-negative parts of f and reflects negative portions above the x-axis.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Write y=|x²−4| piecewise.
3. Solve |x²−4|=5.
3. Challenge
For f(x)=x−2, compare y=|f(x)| with y=f(|x|).
4. Multiple choice
The graph of y=|f(x)| can never have what y-values?
A. positive y-values
B. zero y-values
C. negative y-values
D. a turning point
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Write y=|x−3| piecewise.
y=3−x for x<3; y=x−3 for x≥3.
2. Write y=|x²−4| piecewise.
For |x|<2, x²−4<0 so y=4−x²; for |x|≥2, y=x²−4.
3. Solve |x²−4|=5.
x²−4=5 gives x=±3; x²−4=−5 gives x²=−1, no real roots. Thus x=−3,3.
Challenge. For f(x)=x−2, compare y=|f(x)| with y=f(|x|).
y=|f(x)|=|x−2| is a V with vertex (2,0). y=f(|x|)=|x|−2 is even, has zeros at x=±2 and vertex (0,−2); it reflects the right-hand portion across the y-axis, not negative y-values upward.
Multiple choice. The graph of y=|f(x)| can never have what y-values?
Correct answer: C. Absolute value outputs are always non-negative.
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