Question

UNIT 1: Patterns within Functions

#### 1. Parent Functions, Graphs, Domain, and Range

| Function Type | Equation | Graph Description | Domain | Range |
|-----------------|------------|---------------------|---------|--------|
| Linear | \( y = x \) | Line passing through origin with slope 1 | \(\mathbb{R}\) (all real numbers) | \(\mathbb{R}\) |
| Exponential | \( y = 2^x \) | Curve passing through (0,1), increasing rapidly | \(\mathbb{R}\) | \((0, \infty)\) |
| Quadratic | \( y = x^2 \) | U-shaped parabola opening upward | \(\mathbb{R}\) | \([0, \infty)\) |


#### 2. Linear Function Representation

Equation: \( y = 2x + 3 \)
Table:

| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 |
|--------|--------|--------|---|---|---|
| \( y \) | -1 | 1 | 3 | 5 | 7 |

Graph: A straight line crossing the y-axis at 3 and rising 2 units for every increase of 1 in \(x\).

Visual Model: A ruler or straight-line graph.


#### 3. Exponential Function Representation

Equation: \( y = 3 \times 2^x \)
Table:

| \(x\) | -1 | 0 | 1 | 2 |
|--------|-----|-----|-----|-----|
| \( y \) | 1.5 | 3 | 6 | 12 |

Graph: J-shaped curve increasing rapidly as \(x\) increases.

Visual Model: Exponential growth chart.


#### 4. Quadratic Function Representation

Equation: \( y = x^2 - 4x + 3 \)
Table:

| \(x\) | 0 | 1 | 2 | 3 | 4 |
|--------|---|---|---|---|---|
| \( y \) | 3 | 0 | 1 | 0 | 3 |

Graph: Parabola opening upward, vertex at \( (1, -1) \).

Visual Model: Parabolic curve.


#### 5. Rates of Change Comparison

Linear functions: Constant rate of change (slope). For \( y = 2x + 1 \), rate of change is 2.
Exponential functions: Rate of change is proportional to the current value; it accelerates.
Quadratic functions: Rate of change varies; increases or decreases at a non-constant rate (due to the squared term).


UNIT 2: Quadratic Functions

#### 1. General Forms

Standard form: \( y = ax^2 + bx + c \)
Factored form: \( y = a(x - r1)(x - r2) \)
Vertex form: \( y = a(x - h)^2 + k \)

#### 2. Transformation in Vertex Form

\( y = a(x - h)^2 + k \)
- \(h\) shifts the parabola horizontally.
- \(k\) shifts it vertically.
- \(a\) affects the width and orientation (upward/downward).

#### 3. Graphing a Quadratic

Example: \( y = -2x^2 + 4x + 1 \)

Find vertex: \( h = -\frac{b}{2a} = -\frac{4}{-4} = 1 \)
Calculate \(k\): \( y(1) = -2(1)^2 + 4(1) + 1 = -2 + 4 + 1 = 3 \)
Graph: parabola opening downward with vertex at (1, 3).

#### 4. Solving Quadratic Equations

Factoring: \( x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0 \Rightarrow x=2, 3 \)
Graphical: Find x-intercepts where the parabola crosses x-axis.
Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Example: \( 2x^2 + 4x - 6=0 \)

\[
x = \frac{-4 \pm \sqrt{(4)^2 - 4(2)(-6)}}{2(2)} = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4}
\]
\[
x= \frac{-4 \pm 8}{4} \Rightarrow x=-3, 1
\]

#### 5. Quadratic function: \(f(x) = -2x^2 + 9x + 5\)

Graph:
- Vertex: \(h = -\frac{b}{2a} = -\frac{9}{-4} = 2.25\)
- \(k = f(2.25) = -2(2.25)^2 + 9(2.25) + 5 \approx 11.375 \)
- Maximum point at \((2.25, 11.375)\).
Key features:
- Vertex: \((2.25, 11.375)\) (maximum)
- Axis of symmetry: \(x=2.25\)
- \(x\)-intercepts (via quadratic formula): approximately \(x \approx -0.25, 9.25\)
- \(y\)-intercept: \(f(0)=5\)
- Domain: \(\mathbb{R}\)
- Range: \((-\infty, 11.375]\)

Factored form:
\[
f(x) = -2(x - 0.25)(x - 9.25)
\]

#### 6. Real-World Problem

Problem: A ball is thrown upward with a height modeled by \( h(t) = -16t^2 + 32t + 5 \), where \(t\) is time in seconds and \(h(t)\) is height in meters. When does the ball reach its maximum height?

Solution:

Find vertex: \( t = -\frac{b}{2a} = -\frac{32}{2 \times -16} = 1 \) second.
Max height: \( h(1) = -16(1)^2 + 32(1) + 5 = 21 \) meters.


UNIT 3: Rational Exponents

#### 1. Rational Exponent

Definition: An exponent expressed as a fraction \( \frac{m}{n} \), meaning \( a^{m/n} = \sqrt[n]{a^m} \).

#### 2. Radical

Expression involving roots, such as square root (\(\sqrt{}\)), cube root, etc.

#### 3. Convert from Rational Exponent to Radical

\( a^{m/n} = \sqrt[n]{a^m} \)

Example: \( 8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 \)

#### 4. Simplify Square Roots

Example 1: \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \)

Example 2: \( \sqrt{x^4 y^2} = \sqrt{(x^2)^2 (y)^2} = x^2 y \)

#### 5. Simplify Expressions

a) \( \frac{8x^3}{x^2} \)

\( = 8x^{3-2} = 8x \)

b) \( \frac{x^{-4}}{x^2} \)

\(= x^{-4-2} = x^{-6} = \frac{1}{x^6} \)

c) \( \frac{\sqrt{x^4 y^2}}{\sqrt{x y}} \)

Numerator: \( x^2 y \)
Denominator: \( \sqrt{x y} = x^{1/2} y^{1/2} \)
So: \( \frac{x^2 y}{x^{1/2} y^{1/2}} = x^{2 - 1/2} y^{1 - 1/2} = x^{3/2} y^{1/2} \)

d) \( \sqrt[3]{x^5 y^4} \)

\( = x^{5/3} y^{4/3} \)

Answer 1

Thanks — do you want me to (pick one)
- check these notes for errors (I found some) and correct them,
- make practice problems with answers,
- convert into a one-page study sheet,
- or something else?

While you decide, here are the corrections I found:

1) Unit 1 — Quadratic example y = x^2 − 4x + 3
- Table: f(2) is incorrect. Correct values:
x: 0 → f(0)=3; 1 → f(1)=0; 2 → f(2)= −1; 3 → f(3)=0; 4 → f(4)=3.
- Vertex: your listed vertex (1, −1) is wrong. Correct vertex is (2, −1) (h = 2, k = −1).

2) Unit 2 — f(x) = −2x^2 + 9x + 5
- Vertex x-coordinate: h = −b/(2a) = −9/(2·(−2)) = 9/4 = 2.25 (this was correct).
- Vertex y-coordinate: k = f(9/4) = 121/8 = 15.125 (you had ~11.375 — that's incorrect).
So vertex = (9/4, 121/8) = (2.25, 15.125).
- x-intercepts (roots): solve −2x^2 + 9x + 5 = 0 → roots x = −1/2 and x = 5 (you listed approx −0.25 and 9.25 — incorrect).
- Factored form: −(2x + 1)(x − 5) (equivalently −2(x + 1/2)(x − 5)).
- Range: because the parabola opens down, range = (−∞, 121/8] (not up to 11.375).

Other parts look fine (Unit 1 linear/exponential entries; Unit 3 exponent/radical rules; most algebraic simplifications). Note: when simplifying radicals you should state any implicit domain assumptions when using principal roots (e.g., √x means x ≥ 0).

If you want, I can:
- update the whole document with corrections inline,
- produce practice problems and step-by-step solutions based on these units,
- or create a one-page summary/study guide. Which would you like?