Digital SAT Maths -Unit 1 - 1.7 Absolute Value Function- Study Notes- New Syllabus
Digital SAT Maths -Unit 1 – 1.7 Absolute Value Function- Study Notes- New syllabus
Digital SAT Maths -Unit 1 – 1.7 Absolute Value Function- Study Notes – per latest Syllabus.
Key Concepts:
|x| equations
Inequalities
Graph interpretation
Absolute Value Function \( |x| \) Equations
The absolute value of a number is its distance from 0 on the number line.
Distance is always non-negative.
Examples:
\( |5| = 5 \)
\( |-5| = 5 \)F
Key Idea
An equation involving absolute value usually produces two solutions.
\( |A| = b \) means
\( A = b \) or \( A = -b \)
Steps to Solve \( |x| \) Equations
- Isolate the absolute value
- Create two equations
- Solve both
Important Case
If
\( |A| = -b \) (a negative number)
there is no solution because distance cannot be negative.
Example 1:
Solve:
\( |x| = 7 \)
▶️ Answer/Explanation
\( x = 7 \) or \( x = -7 \)
Solution: \( \{-7,\,7\} \)
Example 2:
Solve:
\( |x – 3| = 5 \)
▶️ Answer/Explanation
Two equations:
\( x – 3 = 5 \)
\( x – 3 = -5 \)
\( x = 8 \)
\( x = -2 \)
Solution: \( \{-2,\,8\} \)
Example 3 :
Solve:
\( |2x + 1| = -4 \)
▶️ Answer/Explanation
Absolute value cannot equal a negative number.
Conclusion: No solution.
Inequalities
Absolute value inequalities describe values within a distance or outside a distance from a number.
Remember:
\( |x-a| \) represents the distance between \( x \) and \( a \).
Two Important Cases
Case 1: “Less Than” (AND situation)
\( |A| < b \) means \( -b < A < b \)
This represents values inside an interval.
Case 2: “Greater Than” (OR situation)
\( |A| > b \) means \( A < -b \) or \( A > b \)
This represents values outside an interval.
Important Note
If \( b < 0 \):
- \( |A| < b \) → no solution
- \( |A| > b \) → all real numbers
Example 1 (Inside Interval):
Solve:
\( |x-4| < 3 \)
▶️ Answer/Explanation
\( -3 < x-4 < 3 \)
Add 4:
\( 1 < x < 7 \)
Solution: \( (1,7) \)
Example 2 (Outside Interval):
Solve:
\( |2x+1| \ge 5 \)
▶️ Answer/Explanation
\( 2x+1 \ge 5 \) or \( 2x+1 \le -5 \)
\( 2x \ge 4 \Rightarrow x \ge 2 \)
\( 2x \le -6 \Rightarrow x \le -3 \)
Solution: \( (-\infty,-3] \cup [2,\infty) \)
Example 3 (SAT Context):
A machine operates correctly only if the temperature is within 4°C of 20°C. Write and solve the inequality.
▶️ Answer/Explanation
\( |T-20| \le 4 \)
\( -4 \le T-20 \le 4 \)
Add 20:
\( 16 \le T \le 24 \)
Conclusion: Temperature must be between 16°C and 24°C.
Graph Interpretation
The graph of an absolute value function has a very special shape called a V-shape.
The parent function is:
\( f(x)=|x| \)
It opens upward and has its lowest point at the origin.
Vertex Form
Most SAT questions use:
\( f(x)=a|x-h|+k \)
This tells you important graph information.
- Vertex → \( (h,k) \)
- \( a>0 \) → opens upward
- \( a<0 \) → opens downward
- Large \( |a| \) → steeper graph
Intercept Meaning
- y-intercept → value when \( x=0 \)
- x-intercepts → where the graph crosses the x-axis
DIGITAL SAT Interpretation
The vertex usually represents a minimum cost, minimum distance, or optimal value.
Example 1 (Vertex):
For \( f(x)=|x-3|+2 \), find the vertex.
▶️ Answer/Explanation
Compare to \( a|x-h|+k \):
\( h=3 \), \( k=2 \)
Vertex: \( (3,2) \)
Example 2 (Minimum Value):
A cost function is \( C(x)=|x-5|+10 \). What is the minimum cost?
▶️ Answer/Explanation
The minimum occurs at the vertex.
\( k=10 \)
Minimum cost: 10
Example 3 (Finding x-intercepts):
Find where \( f(x)=|x-2|-4 \) crosses the x-axis.
▶️ Answer/Explanation
Set \( f(x)=0 \):
\( |x-2|-4=0 \)
\( |x-2|=4 \)
\( x-2=4 \) or \( x-2=-4 \)
\( x=6 \) or \( x=-2 \)
x-intercepts: \( (-2,0) \) and \( (6,0) \)