Asked by Lemy
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
All Answers 1
Answered by
GPT-5 mini
AI
Below is a complete portfolio-style response covering Units 1–3. I include definitions, formulas, worked examples, tables, graph descriptions (how to sketch), and short visual-model descriptions. At the end I ask for clarification about four symbolic expressions in Unit 3 that were not legible in your prompt — please paste those 4 expressions using ^ for exponents and / for fractions (for example (8*x^3)^(1/5) or x^(2/3) * x^(1/4)) and I will simplify them step-by-step.
UNIT 1 — Patterns within Functions
1) Parent functions (graph description, domain, range)
- Linear: y = x
- Graph: straight line through origin with slope 1 (45° line).
- Domain: all real numbers, (-∞, ∞).
- Range: all real numbers, (-∞, ∞).
- Exponential: y = 2^x
- Graph: passes through (0,1). As x → -∞, y → 0 (horizontal asymptote y=0). As x → ∞, y grows rapidly.
- Domain: (-∞, ∞).
- Range: (0, ∞).
- Quadratic: y = x^2
- Graph: parabola opening upward with vertex at (0,0).
- Domain: (-∞, ∞).
- Range: [0, ∞).
2) Representing one linear function four ways
Choose: y = 2x + 1.
- Equation: y = 2x + 1.
- Table of values:
x: -2 -1 0 1 2
y: -3 -1 1 3 5
- Graph description/sketch instructions:
- Plot y-intercept (0,1).
- Use slope 2: from (0,1) go right 1, up 2 to (1,3); or draw slope triangle rise=2 run=1.
- Draw a straight line through plotted points.
- Visual model:
- "Slope triangle" repeatedly along the line or using an arrow showing rise 2 over run 1.
- A stage-by-stage table shows that every increase of x by 1 increases y by constant +2.
3) Representing one exponential function four ways
Choose: y = 2^x.
- Equation: y = 2^x.
- Table:
x: -2 -1 0 1 2 3
y: 1/4 1/2 1 2 4 8
- Graph description:
- Point (0,1). For integer x increase, y doubles each time. Approaches 0 as x->-∞.
- Horizontal asymptote y=0.
- Visual model:
- Doubling blocks: start with 1 block at x=0, then 2, 4, 8 blocks as x increases.
- Ratio model: successive values multiply by 2 (geometric growth), not additive.
4) Representing one quadratic function four ways
Choose: y = x^2 - 4x + 3.
- Equation: y = x^2 - 4x + 3.
- Table:
x: 0 1 2 3 4
y: 3 0 -1 0 3
- Graph description:
- Parabola opening up.
- Vertex at x = -b/(2a) = 4/(2)=2. Vertex value y(2) = 2^2 -8 +3 = -1 → vertex (2,-1).
- x-intercepts (roots) when y=0: factor: (x-1)(x-3) so x=1 and x=3.
- Visual model:
- "Area-square" idea: output equals square of horizontal distance adjusted by linear shift; or a table showing symmetry about axis x=2. Draw symmetric points pairs (1,0) and (3,0); (0,3) and (4,3).
5) Comparing rates of change: linear vs quadratic vs exponential
- Linear: constant rate of change (slope). If y = mx + b, difference y(x+1)-y(x) = m (constant). First difference constant.
- Quadratic: rate of change itself changes at a constant rate. For y = ax^2+bx+c, first differences change linearly; second differences are constant (=2a).
Example: y=x^2: values 0,1,4,9,16; first diffs 1,3,5,7; second diffs 2,2,2.
- Exponential: multiplicative rate of change; successive ratios y(x+1)/y(x) constant (the base). Growth/decay is by constant factor, not constant additive change. Example: y=2^x has ratios 2 each step.
UNIT 2 — Quadratic Functions
1) General forms of quadratic functions
- Standard (expanded) form: f(x) = ax^2 + bx + c.
- Factored form: f(x) = a(x - r1)(x - r2) where r1 and r2 are roots (x-intercepts).
- Vertex (completed-square) form: f(x) = a(x - h)^2 + k where vertex is (h, k).
2) Transformations for vertex form f(x) = a(x - h)^2 + k
- a: vertical stretch (>1) or compression (0<|a|<1). If a < 0, reflection across x-axis (parabola opens down).
- h: horizontal shift by h units; (x - h) moves right by h, so vertex x-coordinate = h.
- k: vertical shift up by k (vertex y-coordinate = k).
- Summary: start with y = x^2, scale by a, shift right h, shift up k, reflect if a<0.
3) How to graph a quadratic (steps)
Given f(x) = ax^2 + bx + c:
- Step 1: Find vertex: h = -b/(2a), k = f(h).
- Step 2: Plot vertex (h,k).
- Step 3: Find axis of symmetry x = h.
- Step 4: Find y-intercept: (0, c).
- Step 5: Find x-intercepts: factor or use quadratic formula; plot them.
- Step 6: Plot a couple more points left/right of vertex; draw smooth parabola through points with correct opening (up if a>0).
Example: Graph y = 2(x - 1)^2 + 3
- Vertex (1,3), opens up (a=2), y-int: y when x=0 → 2(1)+3=5 so (0,5); axis x=1. Plot and sketch.
4) How to solve quadratic equations (three methods)
- Factoring example:
Solve x^2 - 5x + 6 = 0.
Factor: (x - 2)(x - 3) = 0 → x = 2 or x = 3.
- Using a graph example:
Solve x^2 - 4x - 5 = 0.
Graph y = x^2 - 4x - 5 and find x-values where y=0 (x-intercepts). Factoring: (x-5)(x+1) so x = 5 or x = -1. Graphically these are intercepts.
- Quadratic formula example:
Solve 2x^2 + 3x - 2 = 0.
Formula: x = [-b ± sqrt(b^2 - 4ac)]/(2a)
Compute: b^2 - 4ac = 9 - 4*(2)*(-2) = 9 +16 = 25.
x = [-3 ± 5]/(4)
So x = (2/4) = 1/2 or x = (-8/4) = -2.
5) Work the given quadratic: f(x) = −2x^2 + 9x + 5
- Vertex:
h = -b/(2a) = -9/(2 * -2) = 9/4 = 2.25
k = f(9/4) = compute:
f(9/4) = -2*(9/4)^2 + 9*(9/4) + 5
= -2*(81/16) + 81/4 + 5
= -162/16 + 324/16 + 80/16
= 242/16 = 121/8 = 15.125
Vertex: (9/4, 121/8) ≈ (2.25, 15.125).
Since a = -2 < 0, this is a maximum.
- Line (axis) of symmetry: x = 9/4.
- x-intercepts (solve −2x^2 + 9x + 5 = 0):
Multiply by -1: 2x^2 - 9x - 5 = 0.
Discriminant: Δ = (-9)^2 - 4(2)(-5) = 81 + 40 = 121, sqrt=11.
x = [9 ± 11]/4 → x = 20/4 = 5 or x = -2/4 = -1/2.
So x-intercepts: (5,0) and (-1/2, 0).
- y-intercept: f(0) = 5 → (0,5).
- Domain: (-∞, ∞).
- Range: Because parabola opens down and maximum at k=121/8, range: (-∞, 121/8].
- Factored form:
We know roots r1=5 and r2=-1/2. Factor with leading coefficient -2:
f(x) = -2(x - 5)(x + 1/2).
(You can also write f(x) = -2(x - 5)(2x + 1)/2 but the simplest exact factored form preserving integer factors is -2(x - 5)(x + 1/2).)
- Summary list of key features:
- Vertex: (9/4, 121/8) (maximum)
- Axis of symmetry: x = 9/4
- x-intercepts: x = -1/2, x = 5
- y-intercept: (0,5)
- Domain: (-∞, ∞)
- Range: (-∞, 121/8]
6) Real-world quadratic problem (projectile)
Problem: A ball is launched from the ground with initial vertical speed 40 ft/s. Height h(t) (in feet) after t seconds is modeled by h(t) = -16t^2 + 40t. How high does the ball go (maximum height) and when?
Solution:
- Vertex time: t = -b/(2a) = -40/(2 * -16) = 40/32 = 5/4 = 1.25 s.
- Max height: h(1.25) = -16*(1.25)^2 + 40*(1.25) = -16*1.5625 + 50 = -25 + 50 = 25 ft.
So the ball reaches a maximum height of 25 ft at t = 1.25 seconds.
(If you wanted time to hit ground, solve -16t^2 + 40t = 0 → t( -16t + 40) = 0 → t = 0 or t = 40/16 = 2.5 s, so lands at 2.5 s.)
UNIT 3 — Rational Exponents
1) What is a rational exponent?
- A rational exponent is an exponent that is a fraction m/n. a^(m/n) means the n-th root of a raised to the m-th power (or vice versa): a^(m/n) = (n√a)^m = n√(a^m).
2) What is a radical?
- A radical is an expression that uses a root symbol (√). The expression n√a is the principal n-th root of a. The symbol √ without an index means the square root (n=2). Vocabulary:
- Index: n in n√a (the degree of the root).
- Radicand: the expression a inside the root.
- Principal root: the nonnegative root for even indices (√).
- Radical expression: any expression containing a root.
3) Converting rational exponent to radical form
- a^(m/n) = n√(a^m) = (n√a)^m.
- Example: a^(3/2) = (sqrt(a))^3 = (√a)^3 = (a^(1/2))^3 = a^(3/2) = sqrt(a^3) = (sqrt(a))^3.
- When simplifying, it's often convenient to remove perfect powers from under the radical first.
4) Simplifying square roots (including variables)
Rules and important note:
- sqrt(x^2) = |x|, because square root yields a nonnegative result; include absolute values for general-variable simplification when you don't assume sign.
- For even exponents inside an even root, take absolute value if the variable might be negative.
Examples:
a) Simplify sqrt(50 x^3).
- Break into perfect square factors: 50 x^3 = 25 * 2 * x^2 * x.
- sqrt(50 x^3) = sqrt(25 * x^2 * 2x) = sqrt(25)*sqrt(x^2)*sqrt(2x) = 5|x| sqrt(2x).
- If x ≥ 0, this is 5x sqrt(2x). In general, write 5|x| sqrt(2x).
b) Simplify sqrt(72 a^5 b^2).
- Factor: 72 = 36 * 2; a^5 = a^4 * a; b^2 is perfect square.
- sqrt(72 a^5 b^2) = sqrt(36 * 2 * a^4 * a * b^2) = sqrt(36)*sqrt(a^4)*sqrt(b^2)*sqrt(2a) = 6 a^2 |b| sqrt(2a).
- If a,b ≥ 0, then 6a^2 b sqrt(2a).
5) Converting rational exponents to radical and simplification examples
- Example: x^(3/4) = 4√(x^3) = (4√x)^3.
- Example: (27x^6)^(1/3) = 3√(27x^6) = 3√(27)*3√(x^6) = 3*3*(x^2) = 9x^2 (assuming x≥0 for principal root).
6) About negative exponents and writing in simplest radical form
- Rule: a^(-m/n) = 1 / a^(m/n) = 1 / (n√(a^m)).
- When simplifying expressions with rational exponents, first combine powers using exponent rules, then convert to radical form and simplify radicals by removing perfect powers.
REQUEST FOR CLARIFICATION (Unit 3 expressions)
At the end of Unit 3 in your prompt you had four expressions that appear corrupted and I cannot be certain what they are (the glyphs for exponents/fractions did not display clearly). Please re-send those four expressions using plain text math notation (use ^ for powers and / for fractions, parentheses where needed). For example:
- (8*x^3)^(1/5)
- x^(1/2) * x^(3/4)
- x^(5/6) / x^(7/6)
- (6*x^(5))^(3/4)
Once you paste the exact four expressions in clear ASCII form I will simplify each to simplest radical form and show every step, including eliminating negative exponents if present.
If you want, I can also produce printable sheets (PDF-style content) with neat graphs (actual plotted images) for the parent functions and the example functions — tell me if you want images or PDF output and how you will receive them.
UNIT 1 — Patterns within Functions
1) Parent functions (graph description, domain, range)
- Linear: y = x
- Graph: straight line through origin with slope 1 (45° line).
- Domain: all real numbers, (-∞, ∞).
- Range: all real numbers, (-∞, ∞).
- Exponential: y = 2^x
- Graph: passes through (0,1). As x → -∞, y → 0 (horizontal asymptote y=0). As x → ∞, y grows rapidly.
- Domain: (-∞, ∞).
- Range: (0, ∞).
- Quadratic: y = x^2
- Graph: parabola opening upward with vertex at (0,0).
- Domain: (-∞, ∞).
- Range: [0, ∞).
2) Representing one linear function four ways
Choose: y = 2x + 1.
- Equation: y = 2x + 1.
- Table of values:
x: -2 -1 0 1 2
y: -3 -1 1 3 5
- Graph description/sketch instructions:
- Plot y-intercept (0,1).
- Use slope 2: from (0,1) go right 1, up 2 to (1,3); or draw slope triangle rise=2 run=1.
- Draw a straight line through plotted points.
- Visual model:
- "Slope triangle" repeatedly along the line or using an arrow showing rise 2 over run 1.
- A stage-by-stage table shows that every increase of x by 1 increases y by constant +2.
3) Representing one exponential function four ways
Choose: y = 2^x.
- Equation: y = 2^x.
- Table:
x: -2 -1 0 1 2 3
y: 1/4 1/2 1 2 4 8
- Graph description:
- Point (0,1). For integer x increase, y doubles each time. Approaches 0 as x->-∞.
- Horizontal asymptote y=0.
- Visual model:
- Doubling blocks: start with 1 block at x=0, then 2, 4, 8 blocks as x increases.
- Ratio model: successive values multiply by 2 (geometric growth), not additive.
4) Representing one quadratic function four ways
Choose: y = x^2 - 4x + 3.
- Equation: y = x^2 - 4x + 3.
- Table:
x: 0 1 2 3 4
y: 3 0 -1 0 3
- Graph description:
- Parabola opening up.
- Vertex at x = -b/(2a) = 4/(2)=2. Vertex value y(2) = 2^2 -8 +3 = -1 → vertex (2,-1).
- x-intercepts (roots) when y=0: factor: (x-1)(x-3) so x=1 and x=3.
- Visual model:
- "Area-square" idea: output equals square of horizontal distance adjusted by linear shift; or a table showing symmetry about axis x=2. Draw symmetric points pairs (1,0) and (3,0); (0,3) and (4,3).
5) Comparing rates of change: linear vs quadratic vs exponential
- Linear: constant rate of change (slope). If y = mx + b, difference y(x+1)-y(x) = m (constant). First difference constant.
- Quadratic: rate of change itself changes at a constant rate. For y = ax^2+bx+c, first differences change linearly; second differences are constant (=2a).
Example: y=x^2: values 0,1,4,9,16; first diffs 1,3,5,7; second diffs 2,2,2.
- Exponential: multiplicative rate of change; successive ratios y(x+1)/y(x) constant (the base). Growth/decay is by constant factor, not constant additive change. Example: y=2^x has ratios 2 each step.
UNIT 2 — Quadratic Functions
1) General forms of quadratic functions
- Standard (expanded) form: f(x) = ax^2 + bx + c.
- Factored form: f(x) = a(x - r1)(x - r2) where r1 and r2 are roots (x-intercepts).
- Vertex (completed-square) form: f(x) = a(x - h)^2 + k where vertex is (h, k).
2) Transformations for vertex form f(x) = a(x - h)^2 + k
- a: vertical stretch (>1) or compression (0<|a|<1). If a < 0, reflection across x-axis (parabola opens down).
- h: horizontal shift by h units; (x - h) moves right by h, so vertex x-coordinate = h.
- k: vertical shift up by k (vertex y-coordinate = k).
- Summary: start with y = x^2, scale by a, shift right h, shift up k, reflect if a<0.
3) How to graph a quadratic (steps)
Given f(x) = ax^2 + bx + c:
- Step 1: Find vertex: h = -b/(2a), k = f(h).
- Step 2: Plot vertex (h,k).
- Step 3: Find axis of symmetry x = h.
- Step 4: Find y-intercept: (0, c).
- Step 5: Find x-intercepts: factor or use quadratic formula; plot them.
- Step 6: Plot a couple more points left/right of vertex; draw smooth parabola through points with correct opening (up if a>0).
Example: Graph y = 2(x - 1)^2 + 3
- Vertex (1,3), opens up (a=2), y-int: y when x=0 → 2(1)+3=5 so (0,5); axis x=1. Plot and sketch.
4) How to solve quadratic equations (three methods)
- Factoring example:
Solve x^2 - 5x + 6 = 0.
Factor: (x - 2)(x - 3) = 0 → x = 2 or x = 3.
- Using a graph example:
Solve x^2 - 4x - 5 = 0.
Graph y = x^2 - 4x - 5 and find x-values where y=0 (x-intercepts). Factoring: (x-5)(x+1) so x = 5 or x = -1. Graphically these are intercepts.
- Quadratic formula example:
Solve 2x^2 + 3x - 2 = 0.
Formula: x = [-b ± sqrt(b^2 - 4ac)]/(2a)
Compute: b^2 - 4ac = 9 - 4*(2)*(-2) = 9 +16 = 25.
x = [-3 ± 5]/(4)
So x = (2/4) = 1/2 or x = (-8/4) = -2.
5) Work the given quadratic: f(x) = −2x^2 + 9x + 5
- Vertex:
h = -b/(2a) = -9/(2 * -2) = 9/4 = 2.25
k = f(9/4) = compute:
f(9/4) = -2*(9/4)^2 + 9*(9/4) + 5
= -2*(81/16) + 81/4 + 5
= -162/16 + 324/16 + 80/16
= 242/16 = 121/8 = 15.125
Vertex: (9/4, 121/8) ≈ (2.25, 15.125).
Since a = -2 < 0, this is a maximum.
- Line (axis) of symmetry: x = 9/4.
- x-intercepts (solve −2x^2 + 9x + 5 = 0):
Multiply by -1: 2x^2 - 9x - 5 = 0.
Discriminant: Δ = (-9)^2 - 4(2)(-5) = 81 + 40 = 121, sqrt=11.
x = [9 ± 11]/4 → x = 20/4 = 5 or x = -2/4 = -1/2.
So x-intercepts: (5,0) and (-1/2, 0).
- y-intercept: f(0) = 5 → (0,5).
- Domain: (-∞, ∞).
- Range: Because parabola opens down and maximum at k=121/8, range: (-∞, 121/8].
- Factored form:
We know roots r1=5 and r2=-1/2. Factor with leading coefficient -2:
f(x) = -2(x - 5)(x + 1/2).
(You can also write f(x) = -2(x - 5)(2x + 1)/2 but the simplest exact factored form preserving integer factors is -2(x - 5)(x + 1/2).)
- Summary list of key features:
- Vertex: (9/4, 121/8) (maximum)
- Axis of symmetry: x = 9/4
- x-intercepts: x = -1/2, x = 5
- y-intercept: (0,5)
- Domain: (-∞, ∞)
- Range: (-∞, 121/8]
6) Real-world quadratic problem (projectile)
Problem: A ball is launched from the ground with initial vertical speed 40 ft/s. Height h(t) (in feet) after t seconds is modeled by h(t) = -16t^2 + 40t. How high does the ball go (maximum height) and when?
Solution:
- Vertex time: t = -b/(2a) = -40/(2 * -16) = 40/32 = 5/4 = 1.25 s.
- Max height: h(1.25) = -16*(1.25)^2 + 40*(1.25) = -16*1.5625 + 50 = -25 + 50 = 25 ft.
So the ball reaches a maximum height of 25 ft at t = 1.25 seconds.
(If you wanted time to hit ground, solve -16t^2 + 40t = 0 → t( -16t + 40) = 0 → t = 0 or t = 40/16 = 2.5 s, so lands at 2.5 s.)
UNIT 3 — Rational Exponents
1) What is a rational exponent?
- A rational exponent is an exponent that is a fraction m/n. a^(m/n) means the n-th root of a raised to the m-th power (or vice versa): a^(m/n) = (n√a)^m = n√(a^m).
2) What is a radical?
- A radical is an expression that uses a root symbol (√). The expression n√a is the principal n-th root of a. The symbol √ without an index means the square root (n=2). Vocabulary:
- Index: n in n√a (the degree of the root).
- Radicand: the expression a inside the root.
- Principal root: the nonnegative root for even indices (√).
- Radical expression: any expression containing a root.
3) Converting rational exponent to radical form
- a^(m/n) = n√(a^m) = (n√a)^m.
- Example: a^(3/2) = (sqrt(a))^3 = (√a)^3 = (a^(1/2))^3 = a^(3/2) = sqrt(a^3) = (sqrt(a))^3.
- When simplifying, it's often convenient to remove perfect powers from under the radical first.
4) Simplifying square roots (including variables)
Rules and important note:
- sqrt(x^2) = |x|, because square root yields a nonnegative result; include absolute values for general-variable simplification when you don't assume sign.
- For even exponents inside an even root, take absolute value if the variable might be negative.
Examples:
a) Simplify sqrt(50 x^3).
- Break into perfect square factors: 50 x^3 = 25 * 2 * x^2 * x.
- sqrt(50 x^3) = sqrt(25 * x^2 * 2x) = sqrt(25)*sqrt(x^2)*sqrt(2x) = 5|x| sqrt(2x).
- If x ≥ 0, this is 5x sqrt(2x). In general, write 5|x| sqrt(2x).
b) Simplify sqrt(72 a^5 b^2).
- Factor: 72 = 36 * 2; a^5 = a^4 * a; b^2 is perfect square.
- sqrt(72 a^5 b^2) = sqrt(36 * 2 * a^4 * a * b^2) = sqrt(36)*sqrt(a^4)*sqrt(b^2)*sqrt(2a) = 6 a^2 |b| sqrt(2a).
- If a,b ≥ 0, then 6a^2 b sqrt(2a).
5) Converting rational exponents to radical and simplification examples
- Example: x^(3/4) = 4√(x^3) = (4√x)^3.
- Example: (27x^6)^(1/3) = 3√(27x^6) = 3√(27)*3√(x^6) = 3*3*(x^2) = 9x^2 (assuming x≥0 for principal root).
6) About negative exponents and writing in simplest radical form
- Rule: a^(-m/n) = 1 / a^(m/n) = 1 / (n√(a^m)).
- When simplifying expressions with rational exponents, first combine powers using exponent rules, then convert to radical form and simplify radicals by removing perfect powers.
REQUEST FOR CLARIFICATION (Unit 3 expressions)
At the end of Unit 3 in your prompt you had four expressions that appear corrupted and I cannot be certain what they are (the glyphs for exponents/fractions did not display clearly). Please re-send those four expressions using plain text math notation (use ^ for powers and / for fractions, parentheses where needed). For example:
- (8*x^3)^(1/5)
- x^(1/2) * x^(3/4)
- x^(5/6) / x^(7/6)
- (6*x^(5))^(3/4)
Once you paste the exact four expressions in clear ASCII form I will simplify each to simplest radical form and show every step, including eliminating negative exponents if present.
If you want, I can also produce printable sheets (PDF-style content) with neat graphs (actual plotted images) for the parent functions and the example functions — tell me if you want images or PDF output and how you will receive them.
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