Asked by Lemy
IM 2 Portfolio Project
Use your workbooks, notes, previous quizzes, Delta Math examples and online/schoology resources to put
together a portfolio showing what you learned this semester.
This portfolio will count as 50% of your midterm exam grade.
Questions to consider: What are the big ideas of the topic? Are there any acronyms or important vocabulary
to include?
You may handwrite or type your portfolio. The medium you use is up to you! (Ex. slides, booklets, video)
You may not screen shot teacher notes, but you may take pictures of your own notes. The portfolio should
include examples, formulas, and anything else you need to demonstrate the following:
UNIT 1 Patterns within Functions
● Given the parent functions for linear, exponential, and quadratic
functions, create a graph and state the domain and range for each.
● Represent any linear function with a table, graph, equation, and visual
model.
● Represent any exponential function with a table, graph, equation, and
visual model.
● Represent any quadratic function with a table, graph, equation, and visual model.
● How do the rates of change for linear, exponential, and quadratic functions compare to each other?
UNIT 2 Quadratic Functions
● Write the general forms of quadratic functions [standard, factored, and vertex form].
● What are the transformations for quadratic functions in vertex form?
● How do you graph a quadratic function? Include an example.
● How do you solve quadratic equations? Include an example of factoring, using a graph, and quadratic
formula.
● Given 𝑓(𝑥) = −2𝑥ଶ + 9𝑥 + 5 o Graph
o State key features: vertex and if it is a maximum or minimum, line of symmetry, x-intercept(s),
y-intercept, domain, and range.
o Write equation in factored form.
● Write a real world problem that can be solved using a quadratic function. Include the solution and how
you arrived at that answer.
Parent Functions
Linear 𝑦 = 𝑥
Exponential 𝑦 = 2௫ Quadratic 𝑦 = 𝑥ଶ
IM 2 Portfolio Project
UNIT 3 Rational Exponents
● What is a rational exponent?
● What is a radical?
● How do you convert from rational exponent to radical form? Include vocabulary.
● How do you simplify square roots? Include at least 2 examples that include variables.
● Simplify the following expressions so that no negative exponents remain. Show all steps in the process,
even if using a calculator.
Write result in simplest
radical form.
(8𝑥ଶ
)
ଵ
ଷ
Write result in simplest
radical form.
𝑥
ହ
ସ ∙ 𝑥
ଷ
ସ
Write result in simplest
radical form.
𝑥
ଵ
ହ
𝑥
ହ
Write result in simplest
radical form.
൬𝑥
ଵ
ଶ൰ଶ
Your semester portfolio should have at least one page per Unit and be neat enough for Ms. Ricketts to
grade what you have put together. Use the bulleted list above as a checklist for what should be on the
portfolio.
However, this is not an exhaustive list. Feel free to include your own narrative that reflects your
understanding and growth with your mathematics practice this semester.
This is an important project! You will make a portfolio at the end of each semester.
This is how it will be scored:
Points/Criteria
/10 Neatness, Composition, and Organization
/30 Unit 1: Complete and Correct
/30 Unit 2: Complete and Correct
/30 Unit 3: Complete and Correc
All Answers 1
Answered by
GPT-5 mini
AI
I can help you build the full portfolio pages (text, worked examples, and formatting suggestions) so you meet everything your teacher wants. Below I’ve prepared a ready-to-use draft for one page per unit that you can copy/print/handwrite into your portfolio. I also flagged a few expressions in Unit 3 that were not clear — please paste them exactly or send a photo so I can simplify those specific problems.
Unit 1 — Patterns within Functions (one page)
1) Parent functions (show a small sketch of each; describe domain & range)
- Linear (parent): y = x
- Graph: straight line through (0,0) slope 1
- Domain: all real numbers (−∞, ∞)
- Range: all real numbers (−∞, ∞)
- Exponential (parent): y = 2^x
- Graph: horizontal asymptote y = 0, passes through (0,1)
- Domain: all real numbers (−∞, ∞)
- Range: (0, ∞)
- Quadratic (parent): y = x^2
- Graph: parabola with vertex at (0,0) opening upward
- Domain: all real numbers (−∞, ∞)
- Range: [0, ∞)
2) Representations — give one full example each (table, equation, graph description, visual model)
- Linear example: y = 2x − 1
- Table: x: −1, 0, 1, 2 → y: −3, −1, 1, 3
- Graph: straight line with slope 2, y-intercept −1
- Visual model: slope triangle rise 2/run 1 between points
- Equation: y = 2x − 1
- Exponential example: y = 3 · 2^x
- Table: x: −1, 0, 1, 2 → y: 1.5, 3, 6, 12
- Graph: grows multiplicatively; asymptote y=0
- Visual model: growth factor ×2 each +1 in x
- Equation: y = 3·2^x
- Quadratic example: y = x^2 − 2x + 1
- Table: x: 0,1,2,3 → y:1,0,1,4
- Graph: parabola with vertex at (1,0)
- Visual model: parabola; show axis of symmetry x=1
- Equation: y = (x−1)^2
3) Compare rates of change
- Linear: constant rate of change (same first difference).
- Quadratic: rate of change itself changes linearly (first differences are not constant; second differences are constant).
- Exponential: multiplicative rate of change (values multiply by a constant factor each equal step in x) — grows faster than polynomial for large x.
Unit 2 — Quadratic Functions (one page)
1) General forms (write each form + explanation)
- Standard form: f(x) = ax^2 + bx + c (easy to expand; coefficients visible)
- Factored form: f(x) = a(x − r1)(x − r2) (roots/intercepts visible)
- Vertex form: f(x) = a(x − h)^2 + k (vertex at (h,k); easy to transform)
2) Transformations for vertex form f(x)=a(x−h)^2 + k
- h > 0 → shift right h; h < 0 → shift left |h|
- k > 0 → shift up k; k < 0 → shift down |k|
- |a| > 1 → vertical stretch; 0<|a|<1 → vertical compression
- a < 0 → reflection across x-axis
- Axis of symmetry: x = h; vertex: (h,k)
3) How to graph a quadratic (steps)
- Put in vertex form or complete the square to find vertex (h,k).
- Plot vertex and axis of symmetry.
- Find y-intercept (set x=0).
- Find x-intercepts (solve for roots; use factoring or quadratic formula).
- Sketch parabola through these points; show opening direction from sign of a.
4) How to solve quadratics — examples
- Factoring:
Example: x^2 + 5x + 6 = 0 → (x+2)(x+3)=0 → x = −2, −3
- Graphing:
Example: graph y = x^2 − 4. Roots at x = ±2; read x where graph crosses x-axis.
- Quadratic formula:
Example: 2x^2 − 3x − 5 = 0
x = [3 ± sqrt(9 + 40)]/(4) = [3 ± sqrt(49)]/4 = [3 ± 7]/4 → x = 10/4 = 5/2 or x = −1
5) Given f(x) = −2x^2 + 9x + 5 — full analysis
- Vertex: h = −b/(2a) = −9/(2·(−2)) = 9/4 = 2.25
k = f(9/4) = −2(9/4)^2 + 9(9/4) + 5 = 121/8 = 15.125
Vertex = (9/4, 121/8)
- Maximum or minimum: a = −2 < 0 → maximum at vertex
- Axis of symmetry: x = 9/4
- y-intercept: f(0) = 5 → (0,5)
- x-intercepts: solve −2x^2 + 9x + 5 = 0
Multiply by −1: 2x^2 − 9x − 5 = 0 → (2x + 1)(x − 5) = 0
→ x = −1/2 and x = 5 → intercepts: (−1/2,0) and (5,0)
- Domain: all real numbers (−∞, ∞)
- Range: y ≤ 121/8 (since max = 121/8)
- Factored form: −2(x − 5)(x + 1/2) (or −(2x + 1)(x − 5))
6) Real-world quadratic problem (short)
- Problem: A rectangular pen uses 100 m of fencing and is divided into two equal rectangular sections by one fence parallel to a side. Let width = x, length = y. Total fencing for 3 verticals + 2 horizontals: 3x + 2y = 100 → y = (100 − 3x)/2.
Area A = x·y = x(100 − 3x)/2 = 50x − 1.5x^2
This is quadratic: A(x) = −1.5x^2 + 50x. Vertex at x = −b/(2a)= −50/(2·(−1.5)) = 50/3 ≈ 16.67 m
Max area = A(50/3) = compute to get maximum area ≈ 833.33 m^2
(Show full evaluation in your portfolio pages.)
Unit 3 — Rational Exponents (one page)
1) Definitions / vocabulary
- Rational exponent: a^{m/n} means the nth root of a raised to the m power: a^{m/n} = (n√a)^m = n√(a^m).
- Radical: an expression that uses a root symbol √[n]{ } (square root when n=2).
- Radicand: value inside the root symbol.
- Even-index roots: require radicand ≥ 0 (in real numbers). Also note |x| when taking even roots of x^2.
2) Converting between forms
- a^{m/n} = (n√a)^m = n√(a^m)
- Example: 8^{2/3} = (3√8)^2 = (2)^2 = 4 (since 3√8 = 2)
3) Simplifying square roots (with variables) — examples
- Example 1: √(72x^2)
72x^2 = 36·2·x^2 → √(36·2·x^2) = 6|x|√2 → if x≥0, 6x√2.
- Example 2: √(18x^4)
18x^4 = 9·2·x^4 → √(9·2·x^4) = 3x^2√2 (x^2 is nonnegative so no absolute needed on x^2)
4) Simplify negative exponents (method)
- Rule: a^{−r} = 1/a^r
- Combine exponent rules: a^{p}·a^{q} = a^{p+q}; (a^p)^q = a^{pq}
5) Example problems with full steps
- Example A: Simplify (8x^3)^{1/3}
(8x^3)^{1/3} = 8^{1/3} · (x^3)^{1/3} = 2 · x = 2x
- Example B: Simplify x^{2/3} · x^{5/6}
Add exponents: 2/3 + 5/6 = 4/6 + 5/6 = 9/6 = 3/2
So result = x^{3/2} = √(x^3) = x·√x (or x√x if x≥0)
6) (If your assignment asks to "write result in simplest radical form" for specific expressions)
- I saw 4 unclear expressions in the instructions. Please paste them exactly (or send a photo). I’ll simplify each step-by-step and include the final radical forms.
Portfolio formatting & checklist (one small page)
- Make at least one full page per Unit (neat).
- Include: definitions, parent functions & graphs, tables, equations, visual models, worked examples, and the special problems your teacher listed (like fully solved f(x)=−2x^2+9x+5).
- For Unit 3: include vocabulary, conversion examples, at least two variable-containing root simplifications, and the four simplification problems (I need exact text).
- Neatness tips: write large headings, box final answers, number steps, label graphs/points, and include units in real-world problem.
- Rubric reminder: Neatness /10, Unit 1 /30, Unit 2 /30, Unit 3 /30 — make sure each item on the original checklist is present.
Next steps — how I can finish this for you
- I can:
a) Produce a typed, printable 3–4 page portfolio (PDF or plain text) with all worked answers and neatly formatted graphs descriptions.
b) Or give a slide-by-slide Google Slides format you can print or present.
- I need you to confirm:
1) Do you want a typed file or plain text you can copy by hand?
2) Please re-send the 4 expressions in Unit 3 exactly as they appear (some characters didn’t come through clearly). If you prefer, take a quick photo of that part of the assignment and upload it.
3) Any preferred examples for the real-world quadratic (projectile? area? profit?)?
Tell me how you want it delivered and paste or upload the four unclear expressions — I’ll finish all steps and give you a ready-to-print portfolio with every requirement included.
Unit 1 — Patterns within Functions (one page)
1) Parent functions (show a small sketch of each; describe domain & range)
- Linear (parent): y = x
- Graph: straight line through (0,0) slope 1
- Domain: all real numbers (−∞, ∞)
- Range: all real numbers (−∞, ∞)
- Exponential (parent): y = 2^x
- Graph: horizontal asymptote y = 0, passes through (0,1)
- Domain: all real numbers (−∞, ∞)
- Range: (0, ∞)
- Quadratic (parent): y = x^2
- Graph: parabola with vertex at (0,0) opening upward
- Domain: all real numbers (−∞, ∞)
- Range: [0, ∞)
2) Representations — give one full example each (table, equation, graph description, visual model)
- Linear example: y = 2x − 1
- Table: x: −1, 0, 1, 2 → y: −3, −1, 1, 3
- Graph: straight line with slope 2, y-intercept −1
- Visual model: slope triangle rise 2/run 1 between points
- Equation: y = 2x − 1
- Exponential example: y = 3 · 2^x
- Table: x: −1, 0, 1, 2 → y: 1.5, 3, 6, 12
- Graph: grows multiplicatively; asymptote y=0
- Visual model: growth factor ×2 each +1 in x
- Equation: y = 3·2^x
- Quadratic example: y = x^2 − 2x + 1
- Table: x: 0,1,2,3 → y:1,0,1,4
- Graph: parabola with vertex at (1,0)
- Visual model: parabola; show axis of symmetry x=1
- Equation: y = (x−1)^2
3) Compare rates of change
- Linear: constant rate of change (same first difference).
- Quadratic: rate of change itself changes linearly (first differences are not constant; second differences are constant).
- Exponential: multiplicative rate of change (values multiply by a constant factor each equal step in x) — grows faster than polynomial for large x.
Unit 2 — Quadratic Functions (one page)
1) General forms (write each form + explanation)
- Standard form: f(x) = ax^2 + bx + c (easy to expand; coefficients visible)
- Factored form: f(x) = a(x − r1)(x − r2) (roots/intercepts visible)
- Vertex form: f(x) = a(x − h)^2 + k (vertex at (h,k); easy to transform)
2) Transformations for vertex form f(x)=a(x−h)^2 + k
- h > 0 → shift right h; h < 0 → shift left |h|
- k > 0 → shift up k; k < 0 → shift down |k|
- |a| > 1 → vertical stretch; 0<|a|<1 → vertical compression
- a < 0 → reflection across x-axis
- Axis of symmetry: x = h; vertex: (h,k)
3) How to graph a quadratic (steps)
- Put in vertex form or complete the square to find vertex (h,k).
- Plot vertex and axis of symmetry.
- Find y-intercept (set x=0).
- Find x-intercepts (solve for roots; use factoring or quadratic formula).
- Sketch parabola through these points; show opening direction from sign of a.
4) How to solve quadratics — examples
- Factoring:
Example: x^2 + 5x + 6 = 0 → (x+2)(x+3)=0 → x = −2, −3
- Graphing:
Example: graph y = x^2 − 4. Roots at x = ±2; read x where graph crosses x-axis.
- Quadratic formula:
Example: 2x^2 − 3x − 5 = 0
x = [3 ± sqrt(9 + 40)]/(4) = [3 ± sqrt(49)]/4 = [3 ± 7]/4 → x = 10/4 = 5/2 or x = −1
5) Given f(x) = −2x^2 + 9x + 5 — full analysis
- Vertex: h = −b/(2a) = −9/(2·(−2)) = 9/4 = 2.25
k = f(9/4) = −2(9/4)^2 + 9(9/4) + 5 = 121/8 = 15.125
Vertex = (9/4, 121/8)
- Maximum or minimum: a = −2 < 0 → maximum at vertex
- Axis of symmetry: x = 9/4
- y-intercept: f(0) = 5 → (0,5)
- x-intercepts: solve −2x^2 + 9x + 5 = 0
Multiply by −1: 2x^2 − 9x − 5 = 0 → (2x + 1)(x − 5) = 0
→ x = −1/2 and x = 5 → intercepts: (−1/2,0) and (5,0)
- Domain: all real numbers (−∞, ∞)
- Range: y ≤ 121/8 (since max = 121/8)
- Factored form: −2(x − 5)(x + 1/2) (or −(2x + 1)(x − 5))
6) Real-world quadratic problem (short)
- Problem: A rectangular pen uses 100 m of fencing and is divided into two equal rectangular sections by one fence parallel to a side. Let width = x, length = y. Total fencing for 3 verticals + 2 horizontals: 3x + 2y = 100 → y = (100 − 3x)/2.
Area A = x·y = x(100 − 3x)/2 = 50x − 1.5x^2
This is quadratic: A(x) = −1.5x^2 + 50x. Vertex at x = −b/(2a)= −50/(2·(−1.5)) = 50/3 ≈ 16.67 m
Max area = A(50/3) = compute to get maximum area ≈ 833.33 m^2
(Show full evaluation in your portfolio pages.)
Unit 3 — Rational Exponents (one page)
1) Definitions / vocabulary
- Rational exponent: a^{m/n} means the nth root of a raised to the m power: a^{m/n} = (n√a)^m = n√(a^m).
- Radical: an expression that uses a root symbol √[n]{ } (square root when n=2).
- Radicand: value inside the root symbol.
- Even-index roots: require radicand ≥ 0 (in real numbers). Also note |x| when taking even roots of x^2.
2) Converting between forms
- a^{m/n} = (n√a)^m = n√(a^m)
- Example: 8^{2/3} = (3√8)^2 = (2)^2 = 4 (since 3√8 = 2)
3) Simplifying square roots (with variables) — examples
- Example 1: √(72x^2)
72x^2 = 36·2·x^2 → √(36·2·x^2) = 6|x|√2 → if x≥0, 6x√2.
- Example 2: √(18x^4)
18x^4 = 9·2·x^4 → √(9·2·x^4) = 3x^2√2 (x^2 is nonnegative so no absolute needed on x^2)
4) Simplify negative exponents (method)
- Rule: a^{−r} = 1/a^r
- Combine exponent rules: a^{p}·a^{q} = a^{p+q}; (a^p)^q = a^{pq}
5) Example problems with full steps
- Example A: Simplify (8x^3)^{1/3}
(8x^3)^{1/3} = 8^{1/3} · (x^3)^{1/3} = 2 · x = 2x
- Example B: Simplify x^{2/3} · x^{5/6}
Add exponents: 2/3 + 5/6 = 4/6 + 5/6 = 9/6 = 3/2
So result = x^{3/2} = √(x^3) = x·√x (or x√x if x≥0)
6) (If your assignment asks to "write result in simplest radical form" for specific expressions)
- I saw 4 unclear expressions in the instructions. Please paste them exactly (or send a photo). I’ll simplify each step-by-step and include the final radical forms.
Portfolio formatting & checklist (one small page)
- Make at least one full page per Unit (neat).
- Include: definitions, parent functions & graphs, tables, equations, visual models, worked examples, and the special problems your teacher listed (like fully solved f(x)=−2x^2+9x+5).
- For Unit 3: include vocabulary, conversion examples, at least two variable-containing root simplifications, and the four simplification problems (I need exact text).
- Neatness tips: write large headings, box final answers, number steps, label graphs/points, and include units in real-world problem.
- Rubric reminder: Neatness /10, Unit 1 /30, Unit 2 /30, Unit 3 /30 — make sure each item on the original checklist is present.
Next steps — how I can finish this for you
- I can:
a) Produce a typed, printable 3–4 page portfolio (PDF or plain text) with all worked answers and neatly formatted graphs descriptions.
b) Or give a slide-by-slide Google Slides format you can print or present.
- I need you to confirm:
1) Do you want a typed file or plain text you can copy by hand?
2) Please re-send the 4 expressions in Unit 3 exactly as they appear (some characters didn’t come through clearly). If you prefer, take a quick photo of that part of the assignment and upload it.
3) Any preferred examples for the real-world quadratic (projectile? area? profit?)?
Tell me how you want it delivered and paste or upload the four unclear expressions — I’ll finish all steps and give you a ready-to-print portfolio with every requirement included.
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