In this lesson · jump to a section
Introduction
Transformations of Linear Functions helps you connect a mathematical rule to a result you can explain and check. Analyze all four forms af(x), f(x)+d, f(x−c), and f(bx) for f(x)=x. Expand the transformed rule to check its slope and intercept. This lesson belongs to coordinate geometry and graphing and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in transformations of linear functions.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Evaluate functions and use y=mx+b.
Previous lesson: Horizontal and Vertical Lines
Key mathematical vocabulary
- slope
- Vertical change divided by nonzero horizontal change.
- intercept
- A point where a graph meets an axis.
- function
- A rule assigning one output to each permitted input.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
Formulas and rules
- y=mx+b
- m=(y₂−y₁)/(x₂−x₁), x₂≠x₁
- y−y₁=m(x−x₁)
Transform the linear parent
Analyze all four forms af(x), f(x)+d, f(x−c), and f(bx) for f(x)=x. Expand the transformed rule to check its slope and intercept.
The parent function f(x)=x is the identity function. Multiplying its output by a changes its slope and may reflect it across the x-axis. Adding d to its output shifts every point up d units.
Replacing x by x−c shifts the graph right c; replacing x by bx changes its input scale. For this particular parent, f(bx)=bx and bf(x)=bx have the same formula, though horizontal and vertical transformations are different ideas for other parent functions. Expand the transformed rule to check the resulting slope and intercept.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Replace the input by x−22: shift right 22.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Compute g(0) in both the transformed and expanded forms.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
For f(x)=x, compare g(x)=12f(x−22)+6 with f. Give the new y-intercept.
- Replace the input by x−22: shift right 22.
- Multiply outputs by 12: vertical stretch by factor 12. Add 6: shift up 6.
- g(x)=12(x−22)+6=12x+-258. At x=0 the y-intercept value is -258. Also f(12x)=12x changes slope; f(x)+d shifts vertically.
Answer: -258
Verify: Compute g(0) in both the transformed and expanded forms.
Avoid this mistake: Reading x−c as a shift left by c.
Example 2 · Basic
For f(x)=x, compare g(x)=13f(x−22)+6 with f. Give the new y-intercept.
- Replace the input by x−22: shift right 22.
- Multiply outputs by 13: vertical stretch by factor 13. Add 6: shift up 6.
- g(x)=13(x−22)+6=13x+-280. At x=0 the y-intercept value is -280. Also f(13x)=13x changes slope; f(x)+d shifts vertically.
Answer: -280
Verify: Compute g(0) in both the transformed and expanded forms.
Avoid this mistake: Assuming f(x)+d changes the slope.
Example 3 · Intermediate
For f(x)=x, compare g(x)=14f(x−22)+6 with f. Give the new y-intercept.
- Replace the input by x−22: shift right 22.
- Multiply outputs by 14: vertical stretch by factor 14. Add 6: shift up 6.
- g(x)=14(x−22)+6=14x+-302. At x=0 the y-intercept value is -302. Also f(14x)=14x changes slope; f(x)+d shifts vertically.
Answer: -302
Verify: Compute g(0) in both the transformed and expanded forms.
Avoid this mistake: Confusing the order of a vertical stretch and a vertical shift.
Example 4 · Advanced
For f(x)=x, compare g(x)=2f(x−23)+6 with f. Give the new y-intercept.
- Replace the input by x−23: shift right 23.
- Multiply outputs by 2: vertical stretch by factor 2. Add 6: shift up 6.
- g(x)=2(x−23)+6=2x+-40. At x=0 the y-intercept value is -40. Also f(2x)=2x changes slope; f(x)+d shifts vertically.
Answer: -40
Verify: Compute g(0) in both the transformed and expanded forms.
Avoid this mistake: Reading x−c as a shift left by c.
Example 5 · Challenge
For f(x)=x, compare g(x)=3f(x−23)+6 with f. Give the new y-intercept.
- Replace the input by x−23: shift right 23.
- Multiply outputs by 3: vertical stretch by factor 3. Add 6: shift up 6.
- g(x)=3(x−23)+6=3x+-63. At x=0 the y-intercept value is -63. Also f(3x)=3x changes slope; f(x)+d shifts vertically.
Answer: -63
Verify: Compute g(0) in both the transformed and expanded forms.
Avoid this mistake: Assuming f(x)+d changes the slope.
Common mistakes and corrections
- Reading x−c as a shift left by c.
- Assuming f(x)+d changes the slope.
- Confusing the order of a vertical stretch and a vertical shift.
Real-world applications
- Adjust a constant-rate price model for a new starting fee. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Compare distance rules with different speeds and delayed starting times. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Analyze all four forms af(x), f(x)+d, f(x−c), and f(bx) for f(x)=x. Expand the transformed rule to check its slope and intercept.
The parent function f(x)=x is the identity function. Multiplying its output by a changes its slope and may reflect it across the x-axis. Adding d to its output shifts every point up d units.
Frequently asked questions
What should I know before transformations of linear functions?
Evaluate functions and use y=mx+b.
How can I check my work?
Compute g(0) in both the transformed and expanded forms. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Reading x−c as a shift left by c. Assuming f(x)+d changes the slope. Confusing the order of a vertical stretch and a vertical shift.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Adjust a constant-rate price model for a new starting fee.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Analyze all four forms af(x), f(x)+d, f(x−c), and f(bx) for f(x)=x. Expand the transformed rule to check its slope and intercept.
- Reading x−c as a shift left by c.
- Assuming f(x)+d changes the slope.
- Confusing the order of a vertical stretch and a vertical shift.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials