Passport to advanced mathematics on the SAT test

Passport to advanced mathematics subscore on the SAT test

Included topics: quadratic equations and functions; graphing quadratic functions; linear and quadratic systems of equations; exponential expressions; graphing exponential functions; polynomial functions and graphs; radical and rational equations and expressions; isolating quantities; function notation

Passport to advanced mathematics subscore is one of the three SAT math test subscores.

Passport to advanced mathematics subscore includes 9 advanced algebra topics: exponential expressions, quadratic equations and functions, graphing quadratic functions, linear and quadratic systems of equations, graphing exponential functions, polynomial functions and graphs, radical and rational equations and expressions, isolating quantities and function notation. Each one of these 9 topics is explained on a different page on this site, see the left menu or the top menu (under passport to advanced mathematics).

The skills tested in passport to advanced mathematics subscore:

  • The skills tested in Passport to advanced mathematics subscore include making calculations with rational exponents and radicals, rewriting a quadratic function in its different forms, solving quadratic equations, interpreting graphs of functions, solving word problems etc.
  • Some questions test your ability to represent a word problem algebraically or solve a word problem for a given function. Representing a word problem will require defining variables, writing functions and solving equations.

About the test:

  • Passport to advanced mathematics subscore questions include multiple choice questions and student produced response questions.
  • The use of a calculator is permitted for part of the questions.
  • The grade of Passport to advanced mathematics subscore is reported on a scale of 1 to 15.

Before learning this subscore, you need to learn the heart of algebra subscore that includes the basic topics needed for understanding this subscore.

Quadratic equations and quadratic functions SAT topic

A quadratic expression (in Latin quadratus means squared) is an expression that has a squared term (a variable multiplied by itself) and is usually written in a standard form ax2+bx+c where a≠0 (x is a variable and a, b and c are constants).

Quadratic functions and equations questions may ask you to rewrite a quadratic function to one of its 3 forms or to solve a quadratic equation using a formula. Rewriting the quadratic function allows to display its different features: the coordinates of the vertex of the parabola and the x intercepts.

Finding the function of the parabola from its graph topic includes questions about finding the standard form, the quadratic form and the vertex form equations using the vertex coordinates and the x and y intercepts coordinates from the graph of the function.

Other questions include two types of word problems with quadratic functions:  calculating the area of a rectangle problems and calculating height versus time problems. The skills on this subject require writing a quadratic function from a word problem and solving a given quadratic function according to a word problem.

The formula sheet for quadratic equations is given below.

quadratic equations and functions formulas sheets

Graphing quadratic functions SAT topic

Graphing quadratic functions topic includes two parts:

Part 1- The features of the graphs of the quadratic functions– x intercepts, y intercept, vertex points, vertical symmetry and width.

Part 2- Three types of transforming the parabola in the xy plane:
Parabola translation– Shifting the graph up, down, to the left or to the right without changing its width, so that the distance that each point moves is the same.
Parabola translation is done by adding or subtracting  a constant c to the function or from the x variable

Parabola stretching– Changing the graph’s width leaving the x intercepts coordinates and the axis of symmetry of the parabolas the same (the distance that each point moves is not the same).
Parabola stretching is done by multiplying the function by a constant:

Parabola reflecting– Reflecting the graph across the x axis or across the y axis.
Parabola reflecting is done by multiplying the function or the x variable by -1.

The worksheet for graphing quadratic functions is listed below.

Linear and quadratic systems of equations SAT topic

Linear and quadratic systems of equations include 2 equations: a linear equation and a quadratic equation:

A linear equation y=mx+b is an algebraic equation in which each term has an exponent of one. It is called linear because it can be graphed as a straight line in the xy plane.

A quadratic equation y=ax2+bx+c is an equation that has a squared term (a variable multiplied by itself) and is graphed as a parabola in the xy plane. Linear and quadratic system can be solved algebraically or graphically.

Graphical solution of linear and quadratic systems of equations is at the intersection point of the line and the parabola. This intersection can happen 0, 1 or 2 times.

Algebraic solution of linear and quadratic systems of equations can be done by substituting an expression for a variable (plugging an expression instead of a variable). This will leave us with one equation with one variable that we can solve.

Exponential expressions SAT topic

Exponential expression includes 3 components: coefficient, base and exponent. For example: 5x3. The exponent is the number of times we multiply x (in the example the exponent is 3, since x3=x*x*x). The base is the variable (in the example the base is x). The coefficient is the number multiplied by the variable (in the example the coefficient is 5, since 5x3=5*x3.

To solve an exponential expressions questions we need to calculate the values of exponential expressions or solve equations with exponents and radicals.

The skills required for exponential expressions topic include dealing with different types of exponents: positive exponents, zero exponents and rational exponents (exponents with fractions) and dealing with radicals (roots). You also need to know how to solve word problems with exponential expressions: population growth and decay problems and compounding interest problems. 

Performing operations with exponential expressions requires:
Changing the base of exponential expression.
Raising an exponential expression to an exponent.
Adding, subtracting multiplying and dividing exponential expressions.
Multiplying polynomial expressions with the FOIL formula.

Performing operations with radicals (roots) requires:
Rewriting radicals as exponents.
Making calculations with exponents and radicals.
Multiplying and dividing radicals.

Solving word problems questions requires:
Writing an exponential expression from a word problem.
Solving a given exponential expression according to a word problem.
Rounding answers to the nearest whole number.
Calculating the net value of the change in the population.
Calculating the interest that is compounded between 2 periods.
Changing the time units (changing the exponent of the expression).

The formula sheet for rational exponents and radicals is listed below.

Graphing exponential functions SAT topic

Exponential function is a function with a positive constant other than 1 raised to an exponent that includes a variable.

The basic form of the exponential function is f(x)=bx (b is the base and x is the exponent).
The base b is always positive (b>0) and not equal to one (b≠1).
For example: the function f(x)=3x is an exponential function where the base is a constant b=3 and the exponent is the variable x.

The y axis intercept of the basic exponential function graph f(x)=bx is equal to 1 for all values of b.

An exponential function slope is always increasing or always decreasing. The slope form depends on the value of the base b. All graphs of exponential functions are curved.

The ends of the exponential function graph: The graph of a basic exponential function f(x)=bx has a horizontal asymptote on one of its ends (positive x axis or negative x axis). The other end of the function approaches infinity. The end behavior depends on the value of the base b.

We can shift an exponential function graph by adding a constant to the function or by multiplying the exponential term by a coefficient, so that the basic function f(x)=bx will become f(x)=a*bx+d.

Polynomial functions and graphs SAT topic

Polynomial as a mathematical expression made up of more than one term, where each term has a form of axn (for constant a and none negative integer n). For example: 2x3.

In polynomial function the input is raised to second power or higher. The degree of a polynomial function is defined as its highest exponent. Quadratic functions are the simplest form of polynomial functions (they have an exponent of 2).
Even degree polynomial function has an even highest exponent (2, 4, 6, etc.).
Odd degree polynomial function has an odd highest exponent greater than 1 (3, 5, 7, etc.).

For example: The function f(x)=x3+3x2-x-3 is a third degree polynomial function (in this function the highest exponent is 3), it is an odd degree function (the highest exponent 3 is odd).

The factored form of a polynomial function shows the x intercepts of the function.

The standard form of the polynomial function shows the y intercept of the function.

The end behavior shows the location of the graph of the function for very small and very large values of x. To know the end behavior of a polynomial function we need to look at its highest exponent (if it is odd or even) and the sign of its coefficient.

The polynomial remainder theorem says that when we divide a polynomial function f(x) by the expression x-a the remainder is f(a). Therefore, to find the remainder we do not need to do the division, we just plug x=a into f(x) and calculate the output.

Radical and rational equations and expressions SAT topic

Solving radicals and rational exponents requires creating equations that are different from the original equations. This may result in getting extraneous solutions, meaning that the solutions to the new equations don’t satisfy the original equations and therefore are not correct.

Rational expressions: 
A rational expression is an expression containing at least one fraction with a variable in the denominator. The variables in the nominator and the denominator can be a quadratic or a higher order polynomial. For example: 3x/(x2-2) is a rational expression with linear variable in the nominator and a quadratic variable in the denominator.

Solving rational expressions includes simplifying, performing 4 operations with fractions (adding, subtracting, dividing, and multiplying) and cancelling common factors. 

Rewriting a rational expression as a quotient and a remainder can be done with 2 methods: long division and grouping the numerator. 

Rational equations: 
A rational equation is an equation containing at least one fraction with a variable in the denominator. For example: 2/(x+2)=1.

Solving rational equations is done by multiplying both sides of the equation by the least common denominator. Since rational expressions contain a variable in the denominator, we need to exclude an extraneous solution for which the denominator equals to zero (we can’t divide by 0).

Radical equations:
Radicals are rational exponents that are written with roots. For example √x. The symbol of a radical is √ and it represents a square root (instead of writing 2√x we write only √x).

A radical equation is an equation in which a variable appears under a radical. For example: √(x+1)=1 is a radical equation and √(4+12)=x is not a radical equation.

Solving radical equation is done by squaring both of its sides, this action cancels the radical sign and results in a linear or quadratic equation that we can solve. Extraneous solution is a solution that we get after solving a squared equation that is not a solution to the original equation. We need to exclude extraneous solutions since they are incorrect.

Isolating quantities SAT topic

Isolating quantities topic is about isolating variables, so we get a variable equal to an expression containing other variables.

Isolated variable is a variable that is written alone on one side of the equation (the second side of the equation contains other variables). For example: In the equation y=mx+b the variable y is isolated.

Insulating versus solving: We can isolate any variable of the equation by rearranging the equation according to the same rules we use to solve it. These are the rules for maintaining equality: we can add or subtract the same value from each side of the equation and we can divide or multiply each side of the equation by a same value.

Note that the isolated variable will be equal to an expression containing other variables and not a numeric solution like after solving an equation.

Isolating quantities questions may ask you to isolate variables in different types of equations, such as linear equations or quadratic equations.

If you are not familiar with solving an equation topic visit the page about linear equations (heart of algebra subscore) before learning this topic.

Function notation SAT topic

A function is a formula that represents relationship between input variable x (also called independent variable) and output variable f(x) (also called dependent variable). For example: f(x)=2x+5.

In function notation we evaluate the function, meaning finding the output of the function f(x) for a given input x. The input can be numeric (a number), an expression or another function.

An example for a function notation with a numeric input: the value of a function f(x)= 2x+5 when x=2 is f(2)=2x+5=2*2+5=9.
An example for a function notation with an  expression input: the value of a function f(x)= 2x+5 when f(x-1) is f(x-1)=2(x-1)+5=2x-2+5=2x+3.
An example for a function notation with a function g(x) as an input in another function f(x): the value of a function f(x)= 2g(x)+5 when g(x)=2x is f(g(x))=2(2x)+5=4x+5.

Representing values in a table instead of evaluating a function:
Instead of evaluating a function using its formula we can receive a table providing pairs of inputs and outputs with or without seeing the formula that stands behind the table.

For example:

x              f(x)
2               9
t+1           2t+7

The formula that is behind the table is f(x)= 2x+5.  

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