SAT Math: Passport to Advanced Math 📝

Welcome to another SAT Math guide! The Passport to Advanced Math portion of the SAT Math exam makes up 16/58 questions, or about 28% of the math section. After mastering this section, you'll be able to analyze more complex equations and solve systems! Let's get started. 🎈

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🗺️ SAT Passport to Advanced Math Topic Areas

College Board has provided a list of skills that may be tested under "Passport to Advanced Math." There are eight in this list, but don't worry, we'll take you through all of these. Here is a checklist of topics pertaining to this portion of the exam:

➗ Operations with Polynomials and Rewriting Expressions

• • These skills involve assessing your ability to add, subtract, and multiply polynomials by checking how well you can combine and perform basic operations (like adding, subtracting, and multiplying) on expressions that involve variables raised to different powers. 💰 Quadratic Equations & Expressions
• • These skills will help you create a mathematical expression that fits a specific situation or problem, and that expression must have the form of a quadratic equation or function. ⤴️ Exponential Functions, Equations, and Expressions and Radicals
• • These skills will show you that exponential functions show how something keeps getting bigger or smaller by a fixed amount each time period. If it gets bigger, it's called "exponential growth," and if it gets smaller, it's called "exponential decay." 🤷🏽‍♀️ Solving Rational Equations
• • These skills will help you work with rational expressions, which are fractions that may have a variable in the bottom part (denominator). This can involve solving problems with these fractions, known as rational equations. ✅ Systems of Equations
• • These skills will show you how solving a system of equations in two variables means finding the values of the variables that make both equations true. One of the equations is a straight line (linear), while the other can be a curve (quadratic or some other nonlinear shape). 🔄 Relationships Between Algebraic and Graphical Representations of Functions
• • These skills will help you understand how properties of a function relate to properties of its graph and vice versa, including x-intercepts, y-intercept, domain, range, maximum, minimum, increasing, decreasing, end behavior, and transformations. 🤔 Function Notation
• • These skills will help you calculate the value of a function using its rule, and if the function represents a situation, you may need to understand what the value of the function means in that context. Sometimes, you might have to interpret a function when an expression like 2x or x + 1 is given instead of just "x." 👩🏽‍💻 Interpreting and Analyzing More Complex Equations in Context
• • You will be required to find connections between a function, its graph, and the situation it represents. You might have to use an equation to see how changing one value affects another. Sometimes, you'll be asked to rearrange an equation to find the value you're interested in. Also, you may need to create or recognize an equation that gives new insights about the situation or its graph.

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➗ Operations with Polynomials and Rewriting Expressions

Let's begin with operations, which is all about adding, subtracting, and multiplying polynomials.

🧠 What You Need to Know: Operations with Polynomials

• • A polynomial is a mathematical expression made up of variables, coefficients, and non-negative whole-number exponents. Variables are letters that represent unknown numbers, coefficients are the numbers that multiply the variables, and exponents indicate how many times the variable is multiplied by itself.- - For example, the expression 3x² - 2x + 5 is a polynomial because it has variables (x), coefficients (3, -2, 5), and non-negative whole number exponents (2, 1, 0).
• • There are four main operations we can perform with polynomials:- - ➕ Addition: Combining two or more polynomials by adding like terms. Like terms are terms that have the same variable raised to the same exponent.- - Example: (3x² + 2x) + (5x² - x) = 8x² + x- - ➖ Subtraction: Subtracting one polynomial from another, which is similar to addition but with the opposite sign for the terms being subtracted.- - Example: (3x² + 2x) - (5x² - x) = -2x² + 3x- - ✖️ Multiplication: Multiplying two polynomials together. To do this, you need to apply the distributive property, which means multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.- - Example: (2x + 3) * (4x - 1) = 8x² + 5x - 3- - ➗ Division: Dividing one polynomial by another. Division of polynomials can get more complex, but for simplicity, just know that it's about finding how many times one polynomial can fit into another.
• • Rewriting expressions involves simplifying or rearranging expressions to make them easier to work with or to find equivalent forms.- - Simplification: 2x + 3x - x = 4x. In this case, we combined like terms (the terms with 'x') to simplify the expression.- - Equivalent Form: (x + 3)(x - 3) = x² - 9. Here, we used the distributive property to expand the expression on the left side and obtain an equivalent form on the right side. Overall, understanding operations with polynomials and rewriting expressions helps us solve various mathematical problems, including algebraic equations and real-world situations involving variables and their relationships.

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💰 Quadratic Functions and Equations

Skill group number two is all about building quadratic functions! Now let's get into what a quadratic is. 👟

🧠 What You Need to Know: Quadratic Functions and Equations

• • A quadratic expression is a mathematical expression that contains a variable (usually represented by "x") raised to the power of 2. It has the form: ax² + bx + c- - Here, a, b, and c are numbers, and they can be positive, negative, or zero. The term with x² (the variable squared) is what makes it a quadratic expression.
• • A quadratic equation is a special type of equation that involves a quadratic expression set equal to zero. It has the form: ax² + bx + c = 0- - The goal when solving a quadratic equation is to find the values of the variable x that make the equation true.

🤓 Applying Your Knowledge: Quadratic Functions and Equations

Quadratic Equation Solving Practice

Let's say we have the quadratic expression 3x² - 7x + 2 and we are tasked with finding the solutions (roots) to its corresponding quadratic equation. 

1. • Set up the quadratic equation by setting the given quadratic expression to zero: 3x² - 7x + 2 = 0
2. • Solve for x by factoring! This allows us to find the values of x that satisfy this equation. In this case, the solutions are x = 1/3 and x = 2. That's the basic idea behind quadratic equations and expressions. They are essential in many areas of mathematics and have various real-world applications, including physics, engineering, and computer graphics.

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⤴️ Exponential Functions, Equations, and Expressions and Radicals

Onto group three of skills in this topic! Let's discuss a few vocabulary terms you should be familiar with.

🧠 What You Need to Know: Exponentials and Radicals

• • An exponential expression is a mathematical expression where a number, called the base, is raised to a power or exponent. It looks like this: baseᵉˣᵖᵒⁿᵉⁿᵗ- - Example**:** 2³, which is read as "2 raised to the power of 3," equals 2 * 2 * 2 = 8.
• • An exponential function is a special type of mathematical function where the input (usually denoted as 'x') is the exponent. The function has a fixed base, and as the input (x) increases, the output grows exponentially. The general form of an exponential function is f(x) = a * bˣ, where 'a' and 'b' are constants.- - Example**:** f(x) = 2ˣ represents an exponential function with base 2.
• • A radical is a symbol that looks like this: √. It represents the operation of finding the root of a number. The most common radical is the square root (√), which finds the number that, when multiplied by itself, gives the value under the radical symbol.
• • A radical expression is a mathematical expression that includes a radical. For instance, √9 is a radical expression that finds the square root of 9, which is 3.- - Example**:** √16 is a radical expression, and it equals 4 because 4 * 4 = 16.
• • A radical equation is an equation that contains at least one radical expression. Solving these equations involves isolating the variable and getting rid of the radical symbol.- - Example**:** An example of a radical equation is √(x + 5) = 3. To solve this equation, you need to find the value of 'x' that makes the expression under the radical equal to 3.

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🤷🏽‍♀️ Solving Rational Equations

Rational equations often feel tough at first, but once you get in the grove, it gets easier! 🕺🏻🪩

🧠 What You Need to Know: Rational Equations

• • Solving rational equations involves finding the values of the variables that make the fraction (ratio) on the left side of the equation equal to the fraction on the right side. These equations involve expressions with variables in the numerator and denominator. The key concepts for solving rational equations are:
• • Identifying the domain: Before solving, check if there are any values of the variables that would make the denominators zero. Such values are not allowed in rational expressions, as division by zero is undefined. Therefore, you need to determine the domain of the equation.
• • Cross-multiplication: To solve the equation, you can use cross-multiplication, which means multiplying the numerator of one fraction by the denominator of the other fraction and vice versa.
• • Simplifying: After cross-multiplication, you'll end up with an equation that might still have variables. Simplify the equation by combining like terms, moving all variable terms to one side, and constants to the other side.
• • Solving for the variable: Once you have a simplified equation, continue solving for the variable by using standard algebraic techniques such as isolating the variable on one side.

🤓 Applying Your Knowledge: Rational Equations

Solving Rational Equations

Solve for x in the equation (x + 3)/(2x) = 1/(x - 2).

🤔 Step 1: Identify the domain The expressions have denominators of 2x and (x - 2). To find the domain, set each denominator equal to zero and solve for x:

• • 2x = 0 --> x = 0
• • x - 2 = 0 --> x = 2 So, x cannot be 0 or 2, as these values would make the denominators zero, leading to undefined expressions.

✖️ Step 2: Cross-multiplication 

Multiply the numerator of the first fraction with the denominator of the second fraction, and vice versa: (x + 3)(x - 2) = 2x.

👩🏽‍💻 Step 3: Simplifying

Expand the left side by distributing: x² - 2x + 3x - 6 = 2x.

Combine like terms: x² + x - 6 = 2x.

💘 Step 4: Solving for the variable

Move all terms to one side to set the equation to zero: x² + x - 2x - 6 = 0.

Simplify further: x² - x - 6 = 0.

🥳 Step 5: Factor the quadratic equation (if possible) 

In this case, the equation can be factored as: (x - 3)(x + 2) = 0.

✅ Step 6: Solve for x

Now, use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero:

• • x - 3 = 0 --> x = 3
• • x + 2 = 0 --> x = -2 👀 Step 7: Check for extraneous solutions

Since we identified earlier that x cannot be 0 or 2, we need to check the solutions we obtained (x = 3 and x = -2) to ensure they are valid within the domain.

The final solution is x = 3, as x = -2 would make the denominator (x - 2) zero.

Therefore, the solution to the rational equation is x = 3. Phew, that was a lot. 

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✅ Systems of Equations

🧠 What You Need to Know: Systems of Equations

Systems of equations are a collection of two or more equations that involve the same variables. The primary objective is to find values for these variables that satisfy all the equations in the system simultaneously. In other words, we are looking for a common solution that makes all the equations true at the same time.

• • Variables: These are the unknowns in the system represented by letters (e.g., x, y, z).
• • Equations: Mathematical expressions that involve the variables and constants (numbers).
• • Solutions: Values for the variables that satisfy all the equations in the system.

🤓 Applying Your Knowledge: Systems of Equations

Solving a 2x2 System of Equations

Let's take a simple system of equations with two variables, x and y:

Equation 1: 2x + y = 8

Equation 2: x - y = 2

In this system, we have two equations involving the variables x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously.

Now, let's go through this step by step.

1. • Step 1: Choose a method to solve the system. There are various methods to solve systems of equations, such as substitution, elimination, or using matrices. For this example, we will use the elimination method.
2. • Step 2: Elimination. The elimination method involves adding or subtracting the equations to eliminate one variable. We aim to get an equation with only one variable, so we can easily solve for it.1. - We'll add Equation 1 and Equation 2: (2x + y) + (x - y) = 8 + 21. - Simplifying the equation: 3x = 10
3. • Step 3: Solve for x. Now, we have an equation with only one variable, x.1. - To solve for x, we divide both sides by 3: x = 10 / 3
4. • **Step 4: Substitute the value of x into one of the original equations.**1. - We'll use Equation 1: 2(10/3) + y = 8
5. • Step 5: Solve for y. Now, we need to solve for y.1. - We first simplify the equation: 20/3 + y = 81. - Next, we isolate y by subtracting 20/3 from both sides: y = 8 - 20/3
6. • Step 6: Simplify y. To get the value of y, we need a common denominator.1. - y = (24/3) - (20/3)1. - y = 4/3
7. • Step 7: Final solution. Now that we have both x and y values, we can state the solution to the system of equations: x = 10/3 and y = 4/3 By substituting these values back into the original equations, we can verify that they satisfy both equations simultaneously.

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🔄 Relationships Between Algebraic and Graphical Representations of Functions

🧠 What You Need to Know: Algebraic vs. Graphical Representations

• • 😎 Functions: A function is a mathematical relationship between two sets of numbers, usually represented by the letters "x" and "y." For each value of "x" (input), there is a corresponding value of "y" (output) determined by the function's rule.
• • ✍🏻 Algebraic Representation: Functions can be expressed algebraically through equations or formulas. In these representations, the relationship between "x" and "y" is defined explicitly.
• • 📈 Graphical Representation: Functions can also be represented graphically on a coordinate plane. The "x" values are plotted on the horizontal axis, and the "y" values are plotted on the vertical axis.
• • 🌟 Whether the function is represented algebraically or graphically, you should be familiar with the following terms:- - The x-intercept is where a graph or a line crosses the x-axis. At the x-intercept, the y-coordinate is always zero (y = 0).- - The y-intercept is where a graph or a line crosses the y-axis. At the y-intercept, the x-coordinate is always zero (x = 0).- - The domain of a function is the set of all possible input values for a function. It captures all of the x-values for which the function is defined/valid. Think of the domain as the "input zone" of the function.- - The range is the set of all possible output values of a function, representing all the y-values a function can produce based on its domain. Think of the range as the "output zone" of the function.- - The minimum value is the lowest point on the graph of a function within a certain domain. When you're looking at a graph, it is the lowest point on the curve.- - The maximum value is the highest point on the graph of a function within a certain domain and visually would be the highest point on the curve.- - Increasing and decreasing function. If you're given a certain interval, can you indicate if the graph is increasing or decreasing?- - End behavior is the behavior of a function as x approaches positive or negative infinity. Think about what happens to the y values as x becomes extremely large (positive infinity) or extremely small (negative infinity).- - Transformations refer to the changes made to the graph of a function by modifying its equation or formula. These changes can shift, stretch, compress, or reflect the graph.

🤓 Applying Your Knowledge: Algebraic vs. Graphical Representations

Let's consider a simple linear function as an example: f(x) = 2x + 1

Algebraic Representation: The function is given as f(x) = 2x + 1. This means that for any input value "x," the output "f(x)" is equal to twice the value of "x" plus one. For instance, if we plug in "x = 3," the algebraic representation gives us:

• • f(3) = 2 * 3 + 1
• • f(3) = 6 + 1
• • f(3) = 7 So, when "x" is 3, "f(x)" is 7.

Graphical Representation: To graph this function, we'll plot points on the coordinate plane. Choose several "x" values, find their corresponding "f(x)" values using the algebraic representation, and plot the points (x, f(x)).

Plotting these points on the coordinate plane and connecting them results in a straight line. This line is the graphical representation of the function f(x) = 2x + 1.

Visually, you can see that as the "x" values increase, the "y" values increase twice as fast (due to the coefficient 2) and are always one unit higher (due to the constant term 1). This relationship is evident in both the algebraic and graphical representations. Changes in "x" directly impact the corresponding "y" values, and this connection between the two representations helps us better understand the behavior of functions.

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🤔 Function Notation

🧠 What You Need to Know: Function Notation

Function notation is a way of representing mathematical relationships between variables. It uses specific symbols to show how one quantity (usually called the input) is related to another quantity (usually called the output). Key concepts in function notation are:

• • Function: A function is a rule that assigns a unique output value to each input value. For example, you might have a function that takes a number as input and returns its square as output.
• • Function Notation: Function notation allows us to express a function using a simple format, typically using the letter "f" (or another letter) to represent the function, followed by parentheses containing the input variable.
• • Input and Output: The value inside the parentheses is the input to the function, and the value that comes out as a result is the output.

A function notation typically looks like this: f(x). Here, "f" is the name of the function, and "x" represents the input variable. The output of the function when the input is "x" is denoted as f(x).

🤓 Applying Your Knowledge: Function Notation

Now, let's look at an example and explain it step by step: Let's define a function "f" that takes an input "x" and returns the value of 2 times "x" plus 3.

1. • First, we declare the function using the notation: f(x) = 2x + 3.1. - "f" is the name of the function.1. - "x" is the input variable, which can be any real number.1. - "2x + 3" represents the rule for calculating the output, which is 2 times the input plus 3.
2. • If we want to find the output (or evaluate the function) for a specific input value, say, "x = 5," we substitute that value into the function:1. - f(5) = 2 * 5 + 31. - f(5) = 10 + 31. - f(5) = 131. - So, when the input "x" is 5, the output (f(x)) is 13.
3. • Similarly, we can find the output for other input values. For instance, if "x = -2":1. - f(-2) = 2 * (-2) + 31. - f(-2) = -4 + 31. - f(-2) = -11. - The output (f(x)) for the input "x = -2" is -1. Function notation helps us express mathematical relationships concisely and provides a clear way to understand how inputs and outputs are related in a function. 

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👩🏽‍💻 Interpreting and Analyzing More Complex Equations in Context

🧠 What You Need to Know: Analyzing Complex Equations

• • Rearranging Equations: Rearranging an equation involves isolating a specific variable to find its value in terms of other known variables. This helps in solving problems and understanding the relationships between different variables.
• • You may have to use an equation provided to you and explain it in the context of the provided situation.
• • You have to know how to create an equation based on a real-life situation explained.

🤓 Applying Your Knowledge: Analyzing Complex Equations

Applying an Equation to Context Practice

Let's consider a simple example of a function and its graph:

Function: The function represents the relationship between the cost (C) of buying a certain number of tickets (n) to a movie. Let's assume the cost of each ticket is $10.

Equation: The equation for this situation is C = 10 * n. Here, 'n' is the number of tickets purchased, and 'C' is the total cost.

Graph: Now, let's plot the graph of this function. We'll use the x-axis to represent the number of tickets (n) and the y-axis to represent the cost (C).

In this example, the function is given by C = 10 * n, which means the cost (C) is directly proportional to the number of tickets (n) purchased. For every additional ticket purchased, the cost increases by $10.

Graphically, you can see that as the number of tickets increases (n), the cost (C) also increases linearly. Each point on the graph represents a specific input-output pair. For example, if you purchase 2 tickets (n = 2), the total cost would be $20 (C = 10 * 2).

Rearranging Equations Practice

Suppose you want to find out how many tickets you can buy with $50. To do this, we rearrange the equation to solve for 'n': C = 10 * n

Divide both sides by 10: C / 10 = n

Now, plug in C = 50: n = 50 / 10

n = 5

So, you can buy 5 tickets with $50.

This example illustrates how to find connections between the function, its graph, and the situation it represents. It also demonstrates how to rearrange the equation to find the value of interest (number of tickets) given a specific cost.

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💫 Closing

Congratulations! You’ve made it to the end of SAT Math - Passport to Advanced Math. Good luck studying for the SAT Math section, we believe in you! 👏

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