Linear Transformations Practice Worksheet
Linear Transformations Practice
TEKS AR.3A • Algebraic Rules, Graphs, & Slope Comparisons
Name: ______________________
Date: ____________
Period: ___________________
Score: _____ / 10
Anchor Notes: Linear Transformation Form \( g(x) = a \cdot f(x - h) + k \) Parent: \( f(x) = x \)
1. Steepness (\(|a|\))
• \(|a| > 1\): Vertical stretch (steeper line)
• \(0 < |a| < 1\): Vertical compression (flatter line)
2. Direction (\(a < 0\))
• Negative sign (\(-a\)): Reflection across x-axis
• Inverts slope to negative (line decreases)
3. Shifts (\(h\) and \(k\))
• \((x - h)\): Shift right • \((x + h)\): Shift left
• \(+ k\): Shift up • \(- k\): Shift down
A Independent Practice: Core Skills
Questions 1–3
- Given the transformed linear function \( h(x) = -4(x - 3) + 5 \), identify each of the 4 distinct transformations applied to the parent function \( f(x) = x \):
Reflection across: ___________________________________
Dilation factor: ______________________________________
Horizontal shift: ___________________________________
Vertical shift: ____________________________________
- The graphs of linear functions \( f \) and \( g \) are shown on the grid. Parent function \( f(x) = x \) passes through \((0,0)\) and \((6,6)\). Transformed function \( g \) passes through \((0,0)\) and \((6,-2)\).
Which function is best represented by the graph of \( g \)?
A \( g(x) = f(x) - 4 \)
B \( g(x) = -\frac{1}{3}f(x) \)
C \( g(x) = -3f(x) \)
D \( g(x) = -\frac{1}{3}f(x + 2) \)
x y f g
- The linear parent function \( f(x) = x \) is reflected across the x-axis, vertically compressed by a factor of \( \frac{2}{5} \), and translated \( 7 \) units upward to produce \( p(x) \).
Write in form \( p(x) = a \cdot f(x) + k \):
Write in slope-intercept form \( p(x) = mx + b \):
Algebra I • TEKS AR.3A Linear Transformations Page 1 of 3 • Continue to Page 2 →
Linear Transformations Practice • Name: __________________________________
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- Which function represents the linear parent function \( f(x) = x \) vertically stretched by a factor of \( 2 \), translated \( 5 \) units to the left, and translated \( 4 \) units down?
A \( j(x) = 2(x + 5) - 4 \)
B \( j(x) = \frac{1}{2}(x - 5) + 4 \)
C \( j(x) = 2(x - 5) - 4 \)
D \( j(x) = \frac{1}{2}(x + 5) - 4 \)
- Function \( k \) is shown on the grid alongside parent function \( f(x) = x \). Line \( k \) passes through \((0, 3)\) and \((2, -1)\).
Slope of \( k \): \( m = \) ______
y-intercept of \( k \): \( b = \) ______
State all transformations that map \( f(x) \) to \( k(x) \):
1. ________________________________________________________________
2. ________________________________________________________________
x y f(x) k(x)
- Complete the comparative analysis table below for each function compared to the parent function \( f(x) = x \):
| Function | Slope (\(m\)) | Direction | Steepness vs. \( f(x) \) | Transformation Description |
|---|
| \( g(x) = -\frac{5}{2}x \) | \( -\frac{5}{2} \) | Decreasing | [ Steeper / Flatter ] | Reflection & _________________ |
| \( h(x) = \frac{1}{4}x - 2 \) | ________ | [ Inc / Dec ] | [ Steeper / Flatter ] | Vert. compression & ___________ |
| \( r(x) = -x + 6 \) | ________ | [ Inc / Dec ] | [ Same / Steeper / Flatter ] | Reflection & _________________ |
B Higher-Level Thinking & Reasoning
Question 7 of 10
7. Error Analysis: A student was asked to describe the transformations from \( f(x) = x \) to \( m(x) = -3(x + 4) - 2 \).
“The graph has a vertical compression by a factor of 3, is translated 4 units right, and is translated 2 units down. The slope remains positive.”
Identify and correct the 3 specific errors in the student's statement:
Correction 1: ______________________________________________________________________________
Correction 2: ______________________________________________________________________________
Correction 3: ______________________________________________________________________________
Algebra I • TEKS AR.3A Linear Transformations Page 2 of 3 • Continue to Page 3 →
Linear Transformations Practice • Name: __________________________________
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8. Synthesis & Justification: A transformed linear function \( q(x) \) has a negative slope, is steeper than parent function \( f(x) = x \), and has a y-intercept of \((0, -5)\).
Part A: Write one possible equation for \( q(x) \) that satisfies all three conditions:
Part B: Conceptual Justification: Explain why a slope such as \( m = -4 \) produces a graph that is steeper than \( f(x) = x \), even though \(-4 < 1\). Reference the concept of absolute value in your reasoning:
Exit Ticket • Quick Mastery Check (AR.3A)
Questions 9 & 10
- Describe the transformations of parent function \( f(x) = x \) that result in \( h(x) = -\frac{1}{2}(x - 6) + 3 \):
A Reflection across x-axis, stretch by 2, translation 6 units left, and 3 units up
B Reflection across x-axis, compression by \(\frac{1}{2}\), translation 6 units right, and 3 units up
C Compression by factor \(\frac{1}{2}\), translation 6 units right, and 3 units down
D Reflection across x-axis, compression by \(\frac{1}{2}\), translation 6 units left, and 3 units down
- The graphs of linear functions \( f \) and \( w \) are shown below. Which function is best represented by graph \( w \)?
x y f w
A \( w(x) = -\frac{1}{2}f(x) \)
B \( w(x) = -2f(x) \)
C \( w(x) = f(x) - 4 \)
D \( w(x) = 2f(-x) - 2 \)
Student Confidence & Metacognitive Reflection:
I need more practice with negative slopes
I can identify stretches vs. compressions
I have mastered linear transformations
Algebra I • TEKS AR.3A Linear Transformations 10 Total Questions • End of Worksheet
Linear Transformations Answer Key
Teacher Answer Key & Guide
TEKS AR.3A • Linear Transformations & Graph Comparisons
Answer Key • 10 Questions Full Worked Solutions
Common Student Misconceptions & Teaching Tips
Horizontal Shift Sign Confusion: Students frequently misinterpret \((x - 3)\) as shifting left 3. Remind students that the formula is \((x - h)\), so subtracting a positive 3 shifts right.
Negative Slope vs. Steepness: Students assume \(-4\) is "smaller" than 1, so the line must be flatter. Emphasize that steepness depends on \(|m|\), while the negative sign only dictates direction.
A Core Skills Worked Solutions
Questions 1–5
- \( h(x) = -4(x - 3) + 5 \) Transformations:
4 pts (1 per blank)
Reflection: Reflection across the x-axis
Dilation: Vertical stretch by factor of 4
Horizontal: Translation 3 units to the right
Vertical: Translation 5 units upward
- Function represented by graph \( g \):
Correct Answer: B
Correct: B • \( g(x) = -\frac{1}{3}f(x) \)
Rationale: The parent function \( f(x) \) passes through \((0,0)\) with slope \( 1 \). Graph \( g \) passes through \((0,0)\) and \((6,-2)\). Its slope is \( m = \frac{-2 - 0}{6 - 0} = -\frac{2}{6} = -\frac{1}{3} \). Because \( g(x) = -\frac{1}{3}x = -\frac{1}{3}f(x) \), this corresponds to a reflection across the x-axis and a vertical compression by a factor of \(\frac{1}{3}\).
- Writing Transformed Equations:
2 pts
In form \( p(x) = a \cdot f(x) + k \): \( p(x) = -\frac{2}{5}f(x) + 7 \)
In slope-intercept form \( mx + b \): \( p(x) = -\frac{2}{5}x + 7 \)
- Stretch 2, left 5, down 4:
Correct Answer: A
Correct: A • \( j(x) = 2(x + 5) - 4 \)
Distractor Analysis: C has \((x - 5)\) (shift right); B and D use \(\frac{1}{2}\) which is a vertical compression rather than a stretch.
- Graph \( k \) Analysis (Points \((0,3)\) and \((2,-1)\)):
3 pts
Slope: \( m = \frac{-1 - 3}{2 - 0} = \frac{-4}{2} = -2 \)
y-intercept: \( b = 3 \) \((0,3)\)
