IB Mathematics AA HL Absolute value graphs Study Notes
IB Mathematics AA HL Absolute value graphs Study Notes
IB Mathematics AA HL Absolute value graphs Study Notes Offer a clear explanation of Absolute value graphs , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Absolute value graphs.
Absolute Value Graphs
Absolute value graphs and their transformations are essential in understanding key concepts in algebra and calculus. The behavior of these graphs and techniques for solving related equations help in analyzing mathematical functions comprehensively.
Key Concepts
- Graph of \( y = |f(x)| \):
- Replace any section of the graph below the x-axis by reflecting it above the x-axis.
- Graph of \( y = f(|x|) \):
- The portion of the graph to the right of the y-axis is reflected over the y-axis, replacing the section on the left.
- Reciprocal Graph (\( y = \frac{1}{f(x)} \)):
- Vertical asymptote: Solve \( f(x) = 0 \).
- As \( x \to \infty \), \( f(x) \to 0 \): Horizontal Asymptote (H.A.) at \( y = 0 \).
- Y-intercept: Occurs at \( x = 0 \), provided \( f(0) \neq 0 \).
- Test values on either side of vertical asymptotes to determine the graph’s shape.
- Solving Absolute Value Equations:
- A useful tactic is to square both sides, which eliminates the absolute value and simplifies the equation.
Guidance, Clarifications, and Syllabus Links
- Absolute value graphs involve symmetry and reflections, which are crucial in understanding their transformations.
- The reciprocal graph behavior is governed by vertical and horizontal asymptotes.
- Squaring both sides in absolute value equations helps handle the non-linear nature effectively, but care must be taken to check for extraneous solutions.
Examples
Example 1: Graphing \( y = |f(x)| \)
- Given \( f(x) = x^2 – 4x – 5 \), sketch \( y = |f(x)| \):
- Step 1: Find the x-intercepts of \( f(x) = 0 \): \( x = -1, 5 \).
- Step 2: Sketch \( f(x) \). Reflect any part of the curve below the x-axis above the x-axis.
Example 2: Graphing \( y = f(|x|) \)
Given \( f(x) = x^2 – 2x \), graph \( y = f(|x|) \):
Step 1: Sketch \( f(x) \) for \( x \geq 0 \).
Step 2: Reflect the portion to the right of the y-axis over the y-axis.
Example 3: Solving an Absolute Value Equation
Solve \( |x – 3| = 5 \):
Case 1: \( x – 3 = 5 \): \( x = 8 \).
Case 2: \( x – 3 = -5 \): \( x = -2 \).
Solution: \( x = -2, 8 \).
IB Mathematics AA SLAbsolute Value Graphs Exam Style Worked Out Questions
Question: [Maximum mark: 8]
A function f is defined by f(x) = \(\frac{2x – 1}{x + 1}\), where x ∈ R, x ≠ -1.
(a) The graph of y = f (x) has a vertical asymptote and a horizontal asymptote.
Write down the equation of
(i) the vertical asymptote;
(ii) the horizontal asymptote.
(b) On the set of axes below, sketch the graph of y = f (x) .
On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
(c) Hence, solve the inequality \(0<\frac{2x – 1}{x + 1} <2.\)
(d) Solve the inequality \(0<\frac{2|x|-1}{|x| + 1} <2.\)
▶️Answer/Explanation
Ans:
(a) (i) x =−1
(ii) y = 2
(b) 
rational function shape with two branches in opposite quadrants, with two correctly positioned asymptotes and asymptotic behaviour shown axes intercepts clearly shown at \(x = \frac{1}{2} and y = -1\)
(c) x > \(\frac{1}{2}\)
Note: Accept correct alternative correct notation, such as \(\left ( \frac{1}{2}, \infty \right ) and ]\frac{1}{2}, \infty [.\)
(d) EITHER
attempts to sketch \(y = \frac{2|x| – 1}{|x|+1}\)
OR
attempts to solve 2|x| – 1 = 0
Note: Award the (M1) if \(x = \frac{1}{2} and x = -\frac{1}{2}\) are identified.
THEN
\(x <-\frac{1}{2} and x >\frac{1}{2}\)
Note: Accept the use of a comma. Condone the use of ‘and’. Accept correct alternative notation.
Question
The graphs of \(y = \left| {x + 1} \right|\) and \(y = \left| {x – 3} \right|\) are shown below.
Let f (x) = \(\left| {\,x + 1\,} \right| – \left| {\,x – 3\,} \right|\).
Draw the graph of y = f (x) on the blank grid below.
[4]
Hence state the value of
(i) \(f'( – 3)\);
(ii) \(f'(2.7)\);
(iii) \(\int_{ – 3}^{ – 2} {f(x)dx} \).[4]
▶️Answer/Explanation
Markscheme
M1A1A1A1
Note: Award M1 for any of the three sections completely correct, A1 for each correct segment of the graph.
[4 marks]
(i) 0 A1
(ii) 2 A1
(iii) finding area of rectangle (M1)
\( – 4\) A1
Note: Award M1A0 for the answer 4.
[4 marks]