Hoja de repaso: Understanding Exponential and Linear Functions

📋 Course Outline

1. Recognizing Linear vs. Exponential
2. Inverse Functions and Graphs
3. Percent Change and Models
4. Exponential and Logarithmic Equations
5. Logarithm Properties
6. Transformations of Graphs
7. Complex Numbers Operations
8. Factoring Techniques
9. Quadratic with Complex Roots
10. Exponential vs. Linear Growth
11. Inverse and Graph Relationships

📖 1. Recognizing Linear vs. Exponential

🔑 Key Concepts & Definitions

• Additive pattern in tables indicating linear behavior: When the values in a data table increase or decrease by a constant amount between successive entries, the pattern is additive, which signifies a linear function.

• Multiplicative pattern in tables indicating exponential behavior: When the ratio of successive values in a data table remains constant, the pattern is multiplicative, indicating an exponential function.

• Constant rate of change for linear functions: A property where the change in the function's output is the same for equal increases in input, reflected as a constant difference in table values.

• Constant growth factor for exponential functions: The fixed ratio by which the function's value multiplies over equal intervals of input, observed as a consistent ratio between successive table entries.

• Distinguishing linear and exponential from data tables: The process involves analyzing the pattern of change; constant differences suggest linearity, while constant ratios suggest exponential behavior.

📝 Essential Points

• To identify whether a function is linear or exponential from tables:

  • Check if the difference between successive values remains constant (additive pattern). If yes, the function exhibits linear behavior.
  • Check if the ratio of successive values remains constant (multiplicative pattern). If yes, it indicates exponential behavior.

• Linear functions have a constant rate of change, meaning each step increases or decreases by the same amount regardless of where you are on the table.

• Exponential functions have a constant growth factor, meaning each step multiplies the previous value by the same factor.

• Recognizing these patterns helps determine whether data follows a linear or exponential model without needing to graph or perform complex calculations.

💡 Key Takeaway

Data tables reveal whether a relationship is linear or exponential by examining consistent differences or ratios; constant differences indicate linearity, while constant ratios point to exponential growth or decay.

📖 2. Inverse Functions and Graphs

🔑 Key Concepts & Definitions

Inverse Function
A function that reverses the effect of the original function, such that applying the inverse to the output of the original function retrieves the original input. Formally, if $f$ is a function, its inverse $f^{-1}$ satisfies $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$.

Finding Inverse of Linear Functions
The process involves solving for $x$ in terms of $y$ when given a linear function $f(x) = mx + b$. The inverse is obtained by swapping $x$ and $y$, then solving for $y$.

Finding Inverse of Exponential Functions Using Logarithms
This involves rewriting an exponential function in terms of logarithms to solve for the variable. When an exponential function is expressed as $f(x) = A \cdot b^{x}$, its inverse can be found by taking logarithms (e.g., natural log or log base 10), which allows solving for $x$.

Graphical Relationship Between a Function and Its Inverse as Reflection Across y = x
The graph of an inverse function is a reflection of the graph of the original function across the line $y = x$. This means points $(a, b)$ on the graph of $f$ correspond to points $(b, a)$ on the graph of $f^{-1}$.

📝 Essential Points

• The inverse function undoes what the original function does; it "reverses" inputs and outputs.
• To find an inverse algebraically:
  • For linear functions: solve for $x$ in terms of $y$, then replace variables to express as a function.
  • For exponential functions: rewrite as logarithmic equations to isolate the variable.
• The graphical relationship between a function and its inverse is a reflection across the line $y = x$. This reflection property helps visualize inverse functions without algebraic manipulation.
• The domain of a function becomes the range of its inverse, and vice versa.
• Interpreting inverse values in context involves understanding how reversing inputs and outputs relates to real-world scenarios, such as solving for original quantities given their transformed or processed values.

💡 Key Takeaway

The inverse of a function reverses its input-output relationship, and graphically, it appears as a mirror image across the line $y = x$. Finding inverses involves algebraic manipulation—solving for variables using logarithms for exponential functions—and understanding this relationship aids in interpreting inverse values within real contexts.

📖 3. Percent Change and Models

🔑 Key Concepts & Definitions

Percent increase or decrease: The measure of how much a quantity has grown or shrunk relative to its initial value, expressed as a percentage. It is calculated by comparing the difference between the final and initial values, divided by the initial value, then multiplied by 100.

Growth factor over multi-year intervals: The ratio representing how much a quantity multiplies over a specified multi-year period. It is determined by dividing the final value by the initial value over that interval.

Converting multi-year growth factors to per-year factors: The process of finding an equivalent annual growth factor from a multi-year growth factor. This involves taking the root corresponding to the number of years in the interval, effectively finding the per-year multiplicative rate that compounds over multiple years.

Writing exponential models from data (f(t) = A · b^t): Developing a mathematical expression that models data exhibiting exponential behavior, where A is the initial amount, b is the growth (or decay) factor per unit time, and t is time. This model captures multiplicative change over time.

Using exponential models to predict future values: Applying the exponential model to estimate the value of a quantity at future points in time by substituting specific t values into f(t).

Using exponential models to solve for time using logarithms: Rearranging exponential equations to isolate t involves applying logarithms, which convert multiplicative relationships into additive ones, enabling solving for t when given a target value.

📝 Essential Points

• Percent change quantifies how much a quantity increases or decreases relative to its starting point; it can be positive (increase) or negative (decrease).
• Growth factors over multiple years are obtained by dividing the final value by the initial value for that period.
• To find per-year growth factors from multi-year growth factors, take the root corresponding to the number of years; for example, if a 4-year growth factor is known, the annual factor is its 4th root.
• Exponential models are written in the form f(t) = A · b^t, where A is typically the initial amount at t=0, and b indicates how quantities multiply each year.
• Future values can be predicted by plugging in future t values into the exponential model.
• To find how long it takes for a quantity to reach a certain level using an exponential model, rewrite the equation and apply logarithms to solve for t. This involves converting an exponential equation into a linear form via logs.

💡 Key Takeaway

Understanding how to calculate percent change and convert multi-year growth factors into per-year rates allows you to build and manipulate exponential models effectively for prediction and analysis of growth or decay over time.

📖 4. Exponential and Logarithmic Equations

🔑 Key Concepts & Definitions

• Solving exponential equations using logarithms: The process of isolating the exponential expression and applying logarithms to both sides to solve for the variable. This involves rewriting the equation in a form where the variable appears in an exponent, then taking logarithms to linearize the exponential.

• Solving logarithmic equations using properties of logarithms: The method of manipulating equations involving logs by applying properties such as product, quotient, and power rules to condense or expand logs, enabling the isolation of the variable.

• Checking domain restrictions and extraneous solutions in log equations: Verifying that solutions satisfy the original equation's domain constraints, especially since logarithms are only defined for positive arguments. Solutions that violate these restrictions are extraneous and must be discarded.

• Exact solutions involving logarithms: Precise solutions obtained through algebraic manipulation with logs, often expressed in terms of natural or common logarithms, without decimal approximation unless specified.

• Equations combining linear and exponential terms: Equations where a linear expression in the variable is set equal to an exponential expression involving the same variable, requiring specific techniques such as taking logs or algebraic manipulation to solve.

📝 Essential Points

• To solve exponential equations using logarithms:

  • Isolate the exponential term.
  • Take the natural log (ln) or common log (log) of both sides.
  • Use log properties to simplify.
  • Solve for the variable.

• To solve logarithmic equations:

  • Use properties of logs:
    • Product rule: ln(ab) = ln a + ln b
    • Quotient rule: ln(a/b) = ln a − ln b
    • Power rule: ln(a^k) = k ln a
  • Expand or condense logs as needed.
  • Convert complex log expressions into simpler forms.

• When solving log equations:

  • Always check that arguments of logs are positive.
  • Discard solutions that make any log argument non-positive.

• Exact solutions involve expressing answers with logs; decimal approximations are used only when necessary for interpretation.

• Equations with linear and exponential terms often require:

  • Taking logs if the exponential is isolated.
  • Recognizing patterns that allow rewriting in a solvable form.

💡 Key Takeaway

Mastering the use of logarithms—both for solving exponential equations and simplifying log equations—is essential for accurately finding exact solutions while respecting domain restrictions. Always verify solutions to avoid extraneous answers introduced by algebraic manipulations involving logs.

📖 5. Logarithm Properties

🔑 Key Concepts & Definitions

• Product rule of logarithms: This property states that the logarithm of a product is equal to the sum of the logarithms of its factors. Formally,
  $\log_b (xy) = \log_b x + \log_b y$
  where $b$ is the base, and $x, y > 0$.

• Quotient rule of logarithms: This property states that the logarithm of a quotient is equal to the difference of the logarithms of numerator and denominator. Formally,
  $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
  where $x, y > 0$.

• Power rule of logarithms: This property states that the logarithm of a number raised to an exponent equals the exponent times the logarithm of the base number. Formally,
  $\log_b (x^k) = k \log_b x$
  where $x > 0$, $b > 0$, and $b \neq 1$.

• Expanding a single logarithm into sum/difference: This involves expressing a single logarithmic expression as a sum or difference of multiple logs using product and quotient rules. For example,
  $\ln \left(\frac{7x^2 y^{3/2}}{4z}\right) = \ln 7 + 2 \ln x + \frac{3}{2} \ln y - \ln 4 - \ln z$

• Condensing sum/difference of logarithms into a single logarithm: This process combines multiple logs into one by applying the inverse properties—product rule for sums and quotient rule for differences. For example,
  $3 \ln x - \frac{1}{2} \ln y + \ln(6) = \ln(x^3) - \ln(y^{1/2}) + \ln(6) = \ln\left(\frac{6x^3}{y^{1/2}}\right)$

• Domain restrictions for logarithmic expressions: Since logs are only defined for positive arguments, any expression involving logs must satisfy conditions such as $x > 0$, $y > 0$, etc., to be valid.

📝 Essential Points

• The product rule allows splitting or combining logs when dealing with products inside a log.
• The quotient rule is used when logs involve division inside their argument.
• The power rule simplifies logs involving exponents by bringing down the exponent as a coefficient.
• Expanding logs into sums/differences helps in simplifying complex expressions or solving equations.
• Condensing multiple logs into one simplifies expressions and makes solving equations more straightforward.
• Always check domain restrictions before performing operations involving logs; arguments must be positive.
• These properties are fundamental tools for manipulating and simplifying logarithmic expressions in algebraic contexts.

💡 Key Takeaway

Logarithm properties—product, quotient, and power rules—are essential for expanding and condensing expressions, enabling easier manipulation and solution of equations involving logs while respecting their domain restrictions.

📖 6. Transformations of Graphs

🔑 Key Concepts & Definitions

• Horizontal shifts of exponential and logarithmic graphs: Moving the graph left or right along the x-axis without altering its shape. For exponential functions, a shift to the right by h units results in replacing x with x - h. For logarithmic functions, similarly, replacing x with x - h shifts the graph right by h units.

• Vertical shifts of exponential and logarithmic graphs: Moving the graph up or down along the y-axis. This is achieved by adding or subtracting a constant k to the entire function. For exponential functions, f(x) + k shifts upward by k. For logarithmic functions, log_b(x) + k shifts upward by k.

• Vertical reflections of graphs: Flipping the graph across the x-axis. This is done by multiplying the entire function by -1. For example, -f(x) reflects the graph of f(x) across the x-axis, changing all positive y-values to negative and vice versa.

• Vertical stretches and compressions: Changing the steepness or flatness of a graph vertically. Multiplying the function by a factor a > 1 causes a vertical stretch (making it steeper), while multiplying by a factor between 0 and 1 causes a vertical compression (flattening). For example, a · f(x) with a > 1 stretches; with 0 < a < 1 compresses.

• Determining domain and range after transformations:

  • Horizontal shifts do not affect domain; they only translate it.
  • Vertical shifts do not affect domain but shift range accordingly.
  • Vertical reflections invert the range.
  • Stretching/compressing scale affects how values are spread but often does not change domain/range unless restrictions are introduced.

• Identifying horizontal and vertical asymptotes:

  • Horizontal asymptote for exponential functions typically at y = 0 unless shifted.
  • Vertical asymptote for logarithmic functions occurs where the argument of the log is zero or undefined, often at x = some constant after transformations.

• Sketching transformed exponential and logarithmic graphs:

  • Start from the basic graph.
  • Apply horizontal shifts first (move left/right).
  • Then apply vertical shifts (move up/down).
  • Reflect if necessary (flip across x-axis).
  • Stretch or compress vertically.
  • Mark asymptotes based on transformations.
  • Plot key points including intercepts and points obtained from transformations.

📝 Essential Points

• Horizontal shifts are achieved by replacing x with (x − h) in the original function.
• Vertical shifts involve adding/subtracting a constant to/from the entire function.
• Reflection across x-axis involves multiplying the function by −1.
• Vertical stretch/compression involves multiplying the entire function by a positive constant different from 1.
• After transformations:
  • Domain remains unchanged unless horizontal shift affects restrictions.
  • Range is shifted vertically; reflections invert it.
• Asymptotes are identified based on where functions tend toward infinity or zero:
  • Exponential graphs approach y = 0 as x → −∞ (if unshifted).
  • Logarithmic graphs have vertical asymptotes where their argument equals zero after shifting.
• Sketching involves sequentially applying transformations to basic graphs and marking key features like intercepts and asymptotes.

💡 Key Takeaway

Transformations such as shifts, reflections, stretches, and compressions systematically alter exponential and logarithmic graphs' position and shape; understanding these changes allows accurate sketching and analysis of their behavior after modifications.

📖 7. Complex Numbers Operations

🔑 Key Concepts & Definitions

Complex Number
A number expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = -1.
Example: 3 + 4i

Writing in the form a + bi
Any complex number can be written with a real part (a) and an imaginary part (b), separated by a plus sign, with the imaginary unit i attached to the coefficient b. This standard form facilitates operations such as addition, subtraction, multiplication, and division.

Addition and Subtraction of Complex Numbers

• Addition: Combine real parts and imaginary parts separately.
  (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: Subtract real parts and imaginary parts separately.
  (a + bi) - (c + di) = (a - c) + (b - d)i

Multiplication of Complex Numbers
Use distributive property (FOIL method):
(a + bi)(c + di) = ac + adi + bci + bdi²
Since i² = -1, simplify:
(a + bi)(c + di) = ac + (ad + bc)i - bd
Expressed as: (ac - bd) + (ad + bc)i

Division of Complex Numbers Using Complex Conjugates
To divide (a + bi) by (c + di):

1. Multiply numerator and denominator by the conjugate of the denominator, (c - di), to eliminate the imaginary part in the denominator.
2. Compute:
   $\frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}$
3. Simplify numerator using multiplication rules; denominator simplifies to c² + d², which is real.
4. Write the result as a complex number in standard form.

📝 Essential Points

• Standard Form: Writing complex numbers as a + bi is essential for clarity in operations.
• Addition/Subtraction: Performed component-wise; straightforward and similar to vector addition.
• Multiplication: Involves applying distributive property; results in a new complex number with real and imaginary parts derived from original components.
• Division: Requires multiplying numerator and denominator by the conjugate of the denominator to rationalize it; ensures division results in a complex number in standard form.
• Complex Conjugate: For a complex number a + bi, its conjugate is a - bi. Used to rationalize denominators when dividing.

💡 Key Takeaway

Mastering operations with complex numbers involves writing them in standard form, applying algebraic rules for addition, subtraction, multiplication, and division—particularly using conjugates for division—to perform calculations accurately and efficiently.

📖 8. Factoring Techniques

🔑 Key Concepts & Definitions

Factoring difference of squares
A method used to factor expressions that are the difference between two perfect squares. The general form is:
$a^2 - b^2 = (a - b)(a + b)$
where $a$ and $b$ are algebraic expressions or numbers. This technique leverages the identity that the difference of two squares can be expressed as the product of a sum and a difference.

Factoring trinomials with leading coefficient 1
A process to factor quadratic expressions of the form:
$x^2 + bx + c$
by finding two numbers that multiply to $c$ and add to $b$. The trinomial factors into:
$(x + m)(x + n)$
where $m$ and $n$ are the two numbers satisfying these conditions.

Factoring trinomials with leading coefficient greater than 1
A method used for quadratics of the form:
$ax^2 + bx + c \quad (a > 1)$
which involves finding two numbers that multiply to $a \times c$ and add to $b$. The process often includes splitting the middle term or using factoring by grouping after rewriting the trinomial as:
$ax^2 + bx + c = (dx + e)(fx + g)$
where $d, e, f, g$ are numbers determined through systematic trial or algebraic techniques.

📝 Essential Points

• Difference of squares applies only when both terms are perfect squares and separated by subtraction. It simplifies factoring by recognizing this pattern.
• Factoring quadratics with leading coefficient 1 is straightforward once the pair of numbers that satisfy the sum and product conditions are identified.
• Factoring quadratics with leading coefficient greater than 1 is more complex; it often requires finding two numbers that multiply to $a c$ and add to $b$, then rewriting the middle term accordingly.
• These methods are fundamental in simplifying algebraic expressions, solving quadratic equations, and preparing expressions for further algebraic manipulation.
• The process of factoring is essential for solving equations, simplifying rational expressions, and analyzing polynomial functions.

💡 Key Takeaway

Factoring difference of squares, quadratics with leading coefficient 1, and quadratics with leading coefficient greater than 1 are essential techniques that simplify polynomial expressions by recognizing specific patterns, enabling easier solving and manipulation of algebraic equations.

📖 9. Quadratic with Complex Roots

🔑 Key Concepts & Definitions

Forming quadratic from complex conjugate roots:
Given two complex conjugate roots, $a + bi$ and $a - bi$, the quadratic with these roots can be formed by multiplying the factors $(x - (a + bi))$ and $(x - (a - bi))$. This process results in a quadratic with real coefficients.

Factored form of quadratic with complex roots:
A quadratic expressed as the product of its linear factors corresponding to its roots. For roots $a + bi$ and $a - bi$, the factored form is:
$(x - (a + bi))(x - (a - bi))$

Expanding factored form to standard quadratic form:
The process of multiplying out the factors in the factored form to obtain a quadratic in the form:
$ax^2 + bx + c$
with real coefficients, where $a, b, c$ are real numbers.

📝 Essential Points

• When a quadratic has complex roots, they always occur as conjugate pairs: if one root is $a + bi$, the other must be $a - bi$.
• To find the quadratic from complex conjugate roots, multiply their corresponding linear factors:
  $(x - (a + bi))(x - (a - bi))$
• The expansion involves using the difference of squares pattern:
  $[(x - a) - bi][(x - a) + bi] = (x - a)^2 - (bi)^2$
• Since $i^2 = -1$, this simplifies to:
  $(x - a)^2 + b^2$
• The resulting quadratic in standard form is:
  $x^2 - 2ax + (a^2 + b^2)$

💡 Key Takeaway

To construct or expand quadratics with complex conjugate roots, multiply their linear factors and simplify using the difference of squares pattern, resulting in a quadratic with real coefficients.

📖 10. Exponential vs. Linear Growth

🔑 Key Concepts & Definitions

Linear growth model
A model where the quantity increases by a fixed amount over equal time intervals. This pattern exhibits a constant additive increase, meaning each step adds the same value to the previous one.

Exponential growth model
A model where the quantity increases by a fixed multiplicative factor over equal time intervals. This pattern exhibits a multiplicative increase, meaning each step multiplies the previous value by a consistent growth factor.

Comparing values of linear and exponential models over time
When analyzing both models, linear models add a constant amount, resulting in steady, uniform increases. Exponential models multiply by a constant factor, leading to growth that accelerates over time, often surpassing linear growth significantly as time progresses.

Interpretation of differences between linear and exponential growth
The key difference lies in their long-term behavior: linear growth produces steady, predictable increases, while exponential growth results in rapid acceleration, causing the quantities to diverge increasingly over time. Understanding this helps in predicting future values and assessing long-term trends.

📝 Essential Points

• Recognizing whether data follows a linear or exponential pattern can be done by examining tables of values for additive versus multiplicative behaviors.
• In comparing models over time, exponential growth eventually outpaces linear growth due to its multiplicative nature.
• The interpretation of differences emphasizes that exponential models are suitable for phenomena with accelerating growth or decay, whereas linear models are appropriate for steady, uniform change.
• The models' behaviors are fundamental in understanding real-world processes such as population increase, radioactive decay (see source content), or resource expansion.

💡 Key Takeaway

Linear and exponential growth models differ fundamentally in how they increase over time: linear adds a constant amount each period, while exponential multiplies by a fixed factor, leading to vastly different long-term outcomes.

📖 11. Inverse and Graph Relationships

🔑 Key Concepts & Definitions

Domain and Range of Functions and Their Inverses:

• The domain of a function is the set of all possible input values for which the function is defined.
• The range of a function is the set of all possible output values it can produce.
• For the inverse function, the domain and range are swapped relative to the original function:
  • The domain of the inverse is the range of the original function.
  • The range of the inverse is the domain of the original function.

Reflection of Graphs Across the Line y = x:

• The graph of a function and its inverse are mirror images reflected across the line y = x.
• This reflection relationship visually demonstrates how input and output values are interchanged between a function and its inverse.

Finding Inverse Functions Involving Exponentials and Logarithms:

• To find an inverse involving exponential functions, solve for the variable using logarithms, which are the inverse operations of exponentials.
• Conversely, to find an inverse involving logarithmic functions, rewrite equations to isolate the exponential form, then solve for the variable using exponentials.

Interpreting Inverse Function Graphs in Context:

• The graph of an inverse function shows how outputs relate back to inputs in reverse.
• Understanding this reflection helps interpret real-world scenarios where reversing a process (e.g., decoding or undoing a transformation) is necessary.

📝 Essential Points

• The domain and range relationship between a function and its inverse means that understanding one helps determine the other’s restrictions.
• Graphically, if you reflect a function’s graph across y = x, you obtain its inverse’s graph; this reflection illustrates their reciprocal input-output relationship.
• When finding inverses involving exponentials, apply logarithms to switch from exponential form to linear form in terms of logs, enabling algebraic solution for the variable.
• Similarly, for logarithmic functions, rewrite equations in exponential form to solve for variables or to analyze their graphs in relation to their inverses.

💡 Key Takeaway

The graphs of a function and its inverse are mirror images across y = x, with their domains and ranges swapped; understanding this reflection aids in visualizing and solving problems involving inverse functions, especially those with exponential or logarithmic forms.

📊 Synthesis Tables

Concept	Linear Function	Exponential Function	Key Authors/References
Pattern in data tables	Constant difference (additive)	Constant ratio (multiplicative)	N/A
Rate of change	Constant rate of change	Constant growth factor	N/A
Graph reflection	Not applicable	Reflection across y = x line	N/A
Algebraic inverse process	Solve for x, swap variables	Use logarithms to solve for x	N/A
Graph relationship	Not applicable	Reflection across y = x	N/A

⚠️ Common Pitfalls & Confusions

1. Confusing constant difference with constant ratio; linear functions have constant differences, exponential functions have constant ratios.
2. Assuming all functions with increasing values are exponential; linear functions can also increase but with a constant difference.
3. Forgetting that the inverse of a function is a reflection across y = x, leading to misinterpretation of inverse graphs.
4. Misapplying logarithms: using the wrong log base or forgetting to switch the equation when solving exponential equations.
5. Overlooking domain restrictions when finding inverses, especially for exponential and logarithmic functions.
6. Mistaking percent increase for percent change in decay scenarios.
7. Failing to convert multi-year growth factors into per-year factors before modeling.
8. Ignoring the initial value A when constructing exponential models from data.

✅ Exam Checklist

• Know how to identify linear versus exponential functions from data tables by analyzing differences and ratios.
• Understand the concept of constant rate of change for linear functions and constant growth factor for exponential functions.
• Be able to find the inverse of linear functions algebraically and interpret their graphs as reflections across y = x.
• Know how to find the inverse of exponential functions using logarithms and understand their graphical reflection properties.
• Master calculating percent increase/decrease, growth factors over multiple years, and converting these into per-year growth factors.
• Be able to write exponential models from data points in the form f(t) = A · b^t and use them for future predictions.
• Know how to solve exponential equations involving unknowns using logarithms, including rearranging equations appropriately.
• Understand properties of logarithms: product, quotient, power rules, and their applications in solving equations.
• Be familiar with transformations of graphs—shifts, stretches, reflections—and how they relate to function equations.
• Practice operations with complex numbers: addition, subtraction, multiplication, division, and finding conjugates.
• Review factoring techniques: common factoring, difference of squares, quadratic trinomials, sum/difference of cubes.
• Solve quadratic equations with complex roots using the quadratic formula or completing the square.
• Differentiate between exponential growth and linear growth models in real-world contexts.
• Recognize relationships between inverse functions and their graphs; interpret inverse values within context.