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Introduction
Comparing Quadratic Models helps you connect a mathematical rule to a result you can explain and check. Compare opening, vertex, and output values at the same inputs. A narrower parabola need not be higher at every input. This lesson belongs to quadratic expressions and equations 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 comparing quadratic models.
- 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
Use signed arithmetic, factoring, square roots, and function notation.
Previous lesson: Discriminant
Key mathematical vocabulary
- equation
- A statement that two expressions are equal.
- function
- A rule assigning one output to each permitted input.
- domain
- The set of permitted inputs.
- range
- The set of outputs produced.
- factor
- An expression multiplied by another to form a product.
Formulas and rules
- f(x)=ax²+bx+c, a≠0
- f(x)=a(x−h)²+k
- x=(−b±√(b²−4ac))/(2a)
Quadratic functions and equations
Compare opening, vertex, and output values at the same inputs. A narrower parabola need not be higher at every input.
A quadratic function has the form f(x)=ax²+bx+c with a≠0. Its graph is a parabola. Vertex form f(x)=a(x−h)²+k identifies the vertex (h,k), axis x=h, and opening direction from the sign of a.
Factored form connects roots with linear factors: a(x−r₁)(x−r₂) has zeros r₁ and r₂. Solve real quadratic equations by factoring, square roots, completing the square, or the quadratic formula. The discriminant b²−4ac determines whether there are two, one repeated, or no real roots. A quadratic’s mathematical domain is all real numbers, while its range is bounded by its extreme value.
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. Both parabolas have vertex (12,25) and open upward.
- 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. The output difference is a positive square at the specified nonvertex input.
- 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
Compare f(x)=(x−12)²+25 and g(x)=2(x−12)²+25. Which has the larger output at x=23, and where are they equal?
- Both parabolas have vertex (12,25) and open upward.
- At x=23, f=146 and g=267, so g is larger.
- Subtract: g−f=(x−12)². This is zero only at x=12, and positive elsewhere.
Answer: g is larger; equal only at x = 12
Verify: The output difference is a positive square at the specified nonvertex input.
Avoid this mistake: Dropping the ± when solving a squared equation.
Example 2 · Basic
Compare f(x)=(x−13)²+25 and g(x)=2(x−13)²+25. Which has the larger output at x=24, and where are they equal?
- Both parabolas have vertex (13,25) and open upward.
- At x=24, f=146 and g=267, so g is larger.
- Subtract: g−f=(x−13)². This is zero only at x=13, and positive elsewhere.
Answer: g is larger; equal only at x = 13
Verify: The output difference is a positive square at the specified nonvertex input.
Avoid this mistake: Reading the vertex sign incorrectly from (x−h)².
Example 3 · Intermediate
Compare f(x)=(x−14)²+25 and g(x)=2(x−14)²+25. Which has the larger output at x=25, and where are they equal?
- Both parabolas have vertex (14,25) and open upward.
- At x=25, f=146 and g=267, so g is larger.
- Subtract: g−f=(x−14)². This is zero only at x=14, and positive elsewhere.
Answer: g is larger; equal only at x = 14
Verify: The output difference is a positive square at the specified nonvertex input.
Avoid this mistake: Assuming every parabola crosses the x-axis twice.
Example 4 · Advanced
Compare f(x)=(x−2)²+26 and g(x)=2(x−2)²+26. Which has the larger output at x=13, and where are they equal?
- Both parabolas have vertex (2,26) and open upward.
- At x=13, f=147 and g=268, so g is larger.
- Subtract: g−f=(x−2)². This is zero only at x=2, and positive elsewhere.
Answer: g is larger; equal only at x = 2
Verify: The output difference is a positive square at the specified nonvertex input.
Avoid this mistake: Dropping the ± when solving a squared equation.
Example 5 · Challenge
Compare f(x)=(x−3)²+26 and g(x)=2(x−3)²+26. Which has the larger output at x=14, and where are they equal?
- Both parabolas have vertex (3,26) and open upward.
- At x=14, f=147 and g=268, so g is larger.
- Subtract: g−f=(x−3)². This is zero only at x=3, and positive elsewhere.
Answer: g is larger; equal only at x = 3
Verify: The output difference is a positive square at the specified nonvertex input.
Avoid this mistake: Reading the vertex sign incorrectly from (x−h)².
Common mistakes and corrections
- Dropping the ± when solving a squared equation.
- Reading the vertex sign incorrectly from (x−h)².
- Assuming every parabola crosses the x-axis twice.
Real-world applications
- Model the height of a thrown object over its physical time interval. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Find maximum area or minimum cost from a quadratic model. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Compare opening, vertex, and output values at the same inputs. A narrower parabola need not be higher at every input.
A quadratic function has the form f(x)=ax²+bx+c with a≠0. Its graph is a parabola. Vertex form f(x)=a(x−h)²+k identifies the vertex (h,k), axis x=h, and opening direction from the sign of a.
Frequently asked questions
What should I know before comparing quadratic models?
Use signed arithmetic, factoring, square roots, and function notation.
How can I check my work?
The output difference is a positive square at the specified nonvertex input. 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?
Dropping the ± when solving a squared equation. Reading the vertex sign incorrectly from (x−h)². Assuming every parabola crosses the x-axis twice.
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: Model the height of a thrown object over its physical time interval.
- 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
- Compare opening, vertex, and output values at the same inputs. A narrower parabola need not be higher at every input.
- Dropping the ± when solving a squared equation.
- Reading the vertex sign incorrectly from (x−h)².
- Assuming every parabola crosses the x-axis twice.
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