Big Ideas Math, Algebra 2

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 1, students learn how parent functions are transformed through translations, reflections, vertical stretches, and vertical shrinks. Using linear, quadratic, and absolute value parent functions as examples, students graph transformed functions and describe how changes to the equation shift, flip, or rescale the graph. The lesson also covers combinations of transformations, such as applying a reflection and translation together to a single function.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 1, students learn how to apply transformations — including reflections in the x- and y-axis, horizontal and vertical stretches, and horizontal and vertical shrinks — to linear and absolute value functions. Students practice writing new function rules by multiplying inputs or outputs by specific values, such as replacing f(x) with -f(x) for an x-axis reflection or f(ax) for a horizontal shrink. The lesson builds toward combining multiple transformations to produce a single transformed function.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 1, students learn how to write equations of linear functions using slope-intercept form and point-slope form to model real-life situations such as straight-line depreciation and proportional relationships. Students also explore lines of fit, lines of best fit, and the correlation coefficient to analyze scatter plot data. The lesson builds on prior knowledge of slope and linear equations to solve practical problems involving rates of change and real-world contexts.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 1, students learn how to solve systems of three linear equations in three variables using substitution and elimination, and how to express solutions as ordered triples. Students also explore how the geometric intersection of planes in three-dimensional space determines whether a system has exactly one solution, infinitely many solutions, or no solution. The lesson builds on two-variable systems to develop algebraic strategies for identifying consistent and inconsistent three-variable systems.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 2, students learn how the constants a, h, and k in the vertex form g(x) = a(x − h)² + k determine horizontal and vertical translations, reflections across the x-axis, and vertical or horizontal stretches and shrinks of the parent function f(x) = x². Students practice identifying and describing these transformations by matching equations to parabola graphs and writing equations from graphs. The lesson builds fluency with quadratic functions in vertex form as a foundation for deeper work with parabolas throughout the chapter.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 2, students explore key characteristics of quadratic functions, including the axis of symmetry, vertex, and maximum and minimum values of parabolas. Students learn to graph quadratic functions written in both vertex form and standard form by identifying the axis of symmetry using the formula x = -b/2a and plotting points using symmetry. The lesson also introduces intercept form and connects all three forms to real-life problem solving.

In this Grade 8 Algebra 2 lesson from Big Ideas Math Chapter 2, students explore the focus and directrix of a parabola, learning how every point on a parabola is equidistant from these two elements. Students use the Distance Formula to derive and write standard equations of parabolas with vertical and horizontal axes of symmetry, including forms such as y = (1/4p)x² and x = (1/4p)y². Real-world applications like satellite dishes and spotlights illustrate why the focus is the optimal reflection point, connecting geometric definitions to quadratic functions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 2, students learn how to write quadratic equations using vertex form, intercept form, and systems of three linear equations to model real-life situations. Students practice applying these methods to scenarios such as projectile paths and temperature changes, using given vertices, x-intercepts, and data points to determine the value of a. The lesson also introduces quadratic regression and average rate of change as tools for fitting and interpreting quadratic models from real-world data sets.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 3, students learn to solve quadratic equations in standard form using three methods: graphing x-intercepts of the related function, applying square roots with rationalized denominators, and factoring with the Zero-Product Property. The lesson also introduces key vocabulary including roots of an equation and zeros of a function, and connects graphical analysis to the number of real solutions a quadratic equation can have.

In this Grade 8 Algebra 2 lesson from Big Ideas Math Chapter 3, students learn to define and use the imaginary unit i, write complex numbers in standard form a + bi, and identify the subsets of complex numbers including real, imaginary, and pure imaginary numbers. Students practice finding square roots of negative numbers, adding, subtracting, and multiplying complex numbers, and finding complex solutions to quadratic equations. The lesson builds a foundational understanding of how the complex number system extends the real number line to include imaginary numbers.

In this Grade 8 Algebra 2 lesson from Big Ideas Math Chapter 3, students learn how to complete the square by adding (b/2)² to a quadratic expression x² + bx to form a perfect square trinomial. Students practice solving quadratic equations using square roots and the completing the square method, and apply these skills to write quadratic functions in vertex form.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 3, students learn to apply the Quadratic Formula to solve quadratic equations with two real solutions, one real solution, or imaginary solutions. Students also derive the formula by completing the square on the general standard form equation ax² + bx + c = 0, and analyze the discriminant (b² - 4ac) to determine the number and type of solutions. The lesson connects the Quadratic Formula to previously learned methods including factoring, graphing, and completing the square.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 3, students learn how to solve systems of nonlinear equations using graphing, substitution, and elimination methods. The lesson covers systems that pair a quadratic equation with a linear equation or two quadratic equations, exploring how these graphs can intersect in zero, one, or two points. Students practice identifying solutions analytically and graphically, building on their prior knowledge of linear systems.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 3, students learn how to graph quadratic inequalities in two variables and solve quadratic inequalities in one variable. The lesson covers identifying solution regions by graphing parabolas with dashed or solid curves and using test points to determine which region to shade. Students also apply these skills to real-world contexts, such as determining safe weight loads based on rope diameter using a quadratic inequality model.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn to identify polynomial functions by degree and type — including cubic and quartic — and write them in standard form using leading coefficients and constant terms. Students then graph polynomial functions using tables and analyze end behavior to understand how the graph rises or falls on each side. The lesson builds foundational skills for recognizing the shape and x-intercepts of cubic and quartic polynomial graphs.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn how to add, subtract, and multiply polynomials by combining like terms and applying the distributive property. The lesson also covers using Pascal's Triangle to expand binomials and introduces the concept of cubing a binomial using patterns for coefficients and exponents. Students practice both vertical and horizontal formats for polynomial operations, building fluency with standard form and closed sets.

In this Grade 8 Algebra 2 lesson from Big Ideas Math Chapter 4, students learn how to divide polynomials using two methods: polynomial long division and synthetic division. They practice dividing higher-degree polynomials by binomials of the form x − k and apply the Remainder Theorem to check results. The lesson builds on prior knowledge of long division and connects polynomial factors to the zeros of cubic functions.

In this Grade 8 Algebra 2 lesson from Big Ideas Math Chapter 4, students learn how to factor polynomials completely using techniques including finding a common monomial factor, factoring the sum or difference of two cubes, factoring by grouping, and recognizing polynomials in quadratic form. Students also apply the Factor Theorem to connect a polynomial's factors to its zeros and x-intercepts. The lesson builds on prior knowledge of quadratic factoring and synthetic division to handle higher-degree polynomial expressions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn how to solve polynomial equations by factoring, identify repeated solutions and their multiplicities, and analyze how repeated zeros affect a graph's behavior at the x-axis. The lesson also introduces the Rational Root Theorem and the Irrational Conjugates Theorem as tools for finding solutions of higher-degree polynomial equations. Students practice applying the Zero-Product Property, the Perfect Square Trinomial Pattern, and the Difference of Two Squares Pattern to cubic and quartic equations.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn to apply the Fundamental Theorem of Algebra to determine that an nth-degree polynomial equation has exactly n complex solutions, counting repeated roots accordingly. Students also identify complex conjugate pairs of zeros and use Descartes's Rule of Signs to analyze polynomial functions. The lesson covers cubic and quartic equations, building skills in finding both real and imaginary solutions through graphing, synthetic division, and factoring techniques.

In this Grade 8 lesson from Big Ideas Math Algebra 2, students learn how to describe and write transformations of polynomial functions, including horizontal and vertical translations, reflections over the x- and y-axes, and horizontal and vertical stretches and shrinks. Using cubic and quartic parent functions such as f(x) = x³ and f(x) = x⁴, students practice identifying transformation rules expressed in function notation like f(x − h), f(x) + k, a·f(x), and f(ax). The lesson connects polynomial transformations to previously learned techniques applied to linear, absolute value, and quadratic functions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn how to analyze graphs of polynomial functions by using x-intercepts to sketch curves, applying the Location Principle to identify real zeros, and finding turning points to determine local maximums and local minimums. Students also explore the relationship between zeros, factors, solutions, and x-intercepts, and distinguish between even and odd functions based on graph symmetry.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 4, students learn how to model real-life data using polynomial functions by writing cubic functions from given points and applying finite differences to determine the degree of a polynomial that fits equally-spaced data. Students use x-intercepts and known coordinate points to construct factored-form polynomial equations, then verify polynomial degree by analyzing whether first, second, or higher-order differences of y-values are constant and nonzero. The lesson also covers using graphing calculator regression tools to find best-fit polynomial models and evaluate their validity for real-world applications such as baseball distance data.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 5, students learn how to find nth roots of numbers and evaluate expressions with rational exponents, including how the notation √[n]{a} = a^(1/n) connects radical form to exponential form. Students also practice converting between radical and rational exponent expressions and solving equations using nth roots. The lesson distinguishes between even and odd index roots and introduces key vocabulary such as index of a radical and principal root.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 5, students learn how to apply the properties of rational exponents — including Product of Powers, Quotient of Powers, and Power of a Product — to simplify expressions with fractional and negative exponents. Students also practice writing radical expressions in simplest form using the Properties of Radicals, including combining like radicals and working with conjugates. The lesson bridges integer exponent rules to rational exponents, helping students evaluate and simplify products and quotients of radicals with confidence.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 5, students learn to graph radical functions — including square root and cube root functions — by identifying domains and ranges and analyzing parent functions. Students also practice applying transformations such as horizontal and vertical translations, reflections, and stretches or shrinks to radical function graphs. The lesson concludes with writing rules for transformed radical functions and solving real-world problems using square root models.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 5, students learn how to perform arithmetic operations on functions, including addition, subtraction, multiplication, and division of two functions to create new functions. Students practice finding the sum, difference, product, and quotient of functions such as radical and polynomial expressions, and determine the domain of each resulting function. The lesson also explores how to graphically represent the arithmetic combination of two functions by adding corresponding y-values across a shared domain.

In this Algebra 2 lesson from Big Ideas Math Chapter 5, Grade 8 students learn how to find and verify inverse functions by switching the roles of x and y and solving for the new output. The lesson covers key concepts including inverse operations, domain and range interchange, and how the graph of an inverse function is a reflection of the original function across the line y = x. Students also apply inverse functions to solve real-life problems involving nonlinear functions such as quadratics and cube roots.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn to identify and graph exponential growth and decay functions of the form y = ab^x, distinguishing between growth factors (b > 1) and decay factors (0 < b < 1). Students explore key characteristics of exponential function graphs, including domain, range, y-intercept, and asymptote behavior. The lesson also covers using exponential models to solve real-life problems.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn to define and use the natural base e (Euler's number, approximately 2.71828), exploring how the expression (1 + 1/x)^x approaches e as x increases. Students practice simplifying natural base expressions using properties of exponents and graph natural base exponential functions of the form y = ae^(rx), distinguishing between exponential growth and decay.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn to define and evaluate logarithms, rewrite equations between logarithmic form and exponential form, and apply inverse properties of logarithmic and exponential functions. The lesson also covers graphing logarithmic functions, including identifying key characteristics such as domain, range, x-intercept, and asymptotes. Special cases like the common logarithm and natural logarithm are introduced alongside the foundational definition of the logarithm with base b.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn how to apply horizontal and vertical translations, reflections, and horizontal and vertical stretches or shrinks to the graphs of exponential and logarithmic functions. Using parent functions such as f(x) = 4^x, f(x) = e^x, and f(x) = ln x, students identify how changes in the equation affect the graph's position, shape, domain, range, and asymptote. The lesson also covers writing transformation equations and finding the inverses of transformed exponential and logarithmic functions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn the Product, Quotient, and Power Properties of Logarithms and how these properties parallel the corresponding properties of exponents. Students apply these rules to evaluate logarithms, expand expressions like ln(5x⁷/y), condense expressions such as log 9 + 3 log 2 − log 3, and use the change-of-base formula to compute logarithms in any base.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn how to solve exponential and logarithmic equations using the Property of Equality for Exponential Equations, logarithms, and inverse properties. The lesson covers techniques such as rewriting equations with a common base, taking logarithms of both sides, and checking for extraneous solutions. Students also apply these skills to real-world problems like Newton's Law of Cooling to model and solve for unknown values in exponential contexts.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 6, students learn how to classify data sets using finite differences and common ratios to identify exponential, quadratic, and other function types. Students also practice writing exponential functions of the form y = ab^x from two points and use regression technology to find exponential and logarithmic models that fit real-world data. The lesson additionally introduces Gaussian and logistic functions as extensions of exponential modeling.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 7, students learn to classify direct variation and inverse variation by examining equations and data tables, using the defining relationship y = a/x and the constant of variation. Students practice identifying inverse variation by checking whether the products xy are constant and distinguishing it from direct variation, where the ratios y/x are constant. The lesson also covers writing inverse variation equations from real-world contexts, including applications like Hooke's Law and rectangle dimensions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 7, students learn to graph rational functions by identifying vertical and horizontal asymptotes, plotting branches of hyperbolas, and applying transformations such as vertical stretches and translations to the parent function f(x) = 1/x. Students practice writing equations in the form y = a/(x−h) + k to determine shifts in asymptotes and describe changes in domain and range. The lesson builds on prior knowledge of asymptotes and long division to analyze more complex rational functions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 7, students learn how to simplify, multiply, and divide rational expressions by factoring polynomials, dividing out common factors, and identifying excluded values that make a denominator zero. The lesson covers key concepts including simplified form of a rational expression, domain restrictions, and applying the reciprocal method to divide rational expressions.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 7, students learn how to add and subtract rational expressions with both like and unlike denominators by finding least common denominators and simplifying the results. The lesson also covers rewriting rational expressions, graphing related functions, and simplifying complex fractions. Students practice identifying the domain of sums and differences of rational expressions, building directly on their prior knowledge of operations with numerical fractions.

In this Grade 8 Algebra 2 lesson from Big Ideas Math, students learn to define and work with finite and infinite sequences, write rules for the nth term using sequence notation, and identify patterns in both arithmetic and geometric sequences. The lesson also introduces series and summation notation, including sigma notation, as students practice finding the sum of sequence terms. These foundational concepts prepare students for deeper exploration of arithmetic and geometric sequences later in Chapter 8.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 8, students learn to identify arithmetic sequences by checking for a constant common difference between consecutive terms. They practice writing the nth term rule using the formula aₙ = a₁ + (n−1)d and apply it to find specific terms in a sequence. The lesson also introduces arithmetic series and teaches students how to find the sum of finite arithmetic series using Gauss's pairing method.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 8, students learn to identify geometric sequences by finding the common ratio between consecutive terms and write rules using the nth-term formula a_n = a_1 r^(n-1). Students also practice deriving and applying the finite geometric series sum formula by working with sequences that show exponential growth, exponential decay, and alternating signs. The lesson builds skills in recognizing geometric sequences from their graphs and calculating sums of finite geometric series.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 8, students learn how to find partial sums and the sum of an infinite geometric series using the formula S = a₁ ÷ (1 − r). The lesson covers the condition that a finite sum exists only when the absolute value of the common ratio r is less than 1, and explores how partial sums approach a limiting value as n increases. Students apply these concepts through spreadsheet explorations and graphing to build understanding of convergence in infinite geometric series.

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 8, students learn how to evaluate and write recursive rules for sequences, including arithmetic sequences using the recursive equation a_n = a_{n-1} + d and geometric sequences using a_n = r · a_{n-1}. Students also practice translating between recursive and explicit rules and apply recursive rules to real-life problems. Special sequences such as the Fibonacci sequence and factorial sequences are introduced as examples of recursive patterns that are neither strictly arithmetic nor geometric.