MathChapter 3: Functions and Linear Equations
Session progress0 of 20 studied
- Again
- 0
- Hard
- 0
- Good
- 0
- Easy
- 0
Card 1 of 20. Answer hidden.
Prompt
What does absolute value measure?
💡Distance is nonnegative.
Answer
Distance from zero on a number line.
Prompt
Can an absolute value be negative?
💡Distance cannot be negative.
Answer
No. Absolute value is always zero or positive.
Example\(|x|=-2\) has no solution.
Prompt
Solve |x| = a when a > 0.
💡Two equal-distance points.
Answer
\(x=a\) or \(x=-a\).
Example\(|x|=5\) gives \(x=\pm5\).
Prompt
Solve |x| = 0.
💡Only zero is zero units from zero.
Answer
\(x=0\) only.
ExampleThere is one solution.
Prompt
Solve |x| = a when a < 0.
💡Absolute value is nonnegative.
Answer
There is no real solution.
ExampleDo not create two branches from a negative target.
Prompt
What must happen before splitting an absolute-value equation?
💡Remove outside constants and coefficients.
Answer
Isolate the absolute-value expression.
ExampleTurn \(2|x-1|+3=11\) into \(|x-1|=4\).
Prompt
How do you split |E| = a for positive a?
💡Write both branches.
Answer
Solve \(E=a\) and \(E=-a\).
ExampleCheck both candidates.
Prompt
Why can |E| = 0 not produce two distinct branches?
💡Both branches coincide.
Answer
The positive and negative targets are both zero.
ExampleSolve \(E=0\) once.
Prompt
What is the parent absolute-value function?
💡Its graph is V-shaped.
Answer
\(f(x)=|x|\).
Prompt
State the piecewise definition of |x|.
💡Negative inputs are negated.
Answer
\(|x|=-x\) for \(x<0\), and \(|x|=x\) for \(x\ge0\).
ExampleBoth branches output nonnegative values.
Prompt
What is the vertex of y = a|x-h| + k?
💡The visible horizontal sign is reversed.
Answer
\((h,k)\).
Example\(y=|x+3|-2\) has vertex \((-3,-2)\).
Prompt
What does the sign of a control in y = a|x-h| + k?
💡It reflects across the x-axis when negative.
Answer
Opening direction: up if positive, down if negative.
Example\(-|x|\) opens down.
Prompt
What does |a| control in an absolute-value graph?
💡Larger magnitude is steeper.
Answer
Vertical steepness or stretch/compression.
Example\(2|x|\) is steeper than \(|x|\).
Prompt
What does h control?
Answer
Horizontal shift and the vertex x-coordinate.
Example\(|x-4|\) shifts right 4.
Prompt
What does k control?
💡Its displayed sign is direct.
Answer
Vertical shift and the vertex y-coordinate.
Example\(|x|+3\) shifts up 3.
Prompt
What is the common branch-sign error?
💡Use E = -a, not -E = -a unless equivalent.
Answer
Writing the negative branch incorrectly or forgetting it.
ExampleKeep the full inside expression together.
Prompt
How does absolute difference model distance?
💡Order does not affect absolute difference.
Answer
The distance between x and c is \(|x-c|\).
ExampleExactly 3 from 10: \(|x-10|=3\).
Prompt
Why should candidate solutions be checked?
💡Use the original equation.
Answer
Algebraic sign errors are common, and outside operations may have been mishandled.
ExampleBoth branches must reach the stated absolute value.
Prompt
What is the symmetry shortcut for |x-h| = a?
💡Their average is h.
Answer
The solutions are \(h-a\) and \(h+a\), centered at h.
ExampleTheir sum is \(2h\).
Prompt
Does every absolute-value equation have two solutions?
💡Inspect after isolation.
Answer
No: target sign and transformations determine whether there are zero, one, or two.
Example\(|x|=0\) has one; \(|x|=-1\) has none.
✓Deck completeYou rated every card. Revisit difficult cards or shuffle for another pass.
Keyboard: Space reveals, left/right arrows navigate, 1–4 rate, and S shuffles.