Grade 11 · secular · Common Core-aligned
This is the algebra course that takes "graphing parabolas" from Algebra I and blows it out into six more families of functions, polynomials, rational expressions, radicals, exponentials/logs, trig, and finally sequences and statistics. The thread running through all eight units is one question asked over and over in new clothes: given an equation, what does the graph look like, and given a graph or a real situation, what's the equation? Your child will get a lot of practice moving between an equation, a graph, a table, and a word problem, for each of these families, and by the end they should be able to look at data or a situation and say "that's exponential, not linear" or "that needs a hole, not an asymptote" and explain why. It's a full year of building one increasingly large toolbox rather than eight unrelated topics.
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This unit takes quadratics, which your child can already factor and graph from Algebra I, and gives them three ways to write the same equation (standard, factored, vertex), a way to predict how many solutions an equation has before solving it (the discriminant), and a brand-new kind of number for when the answer to a square root is negative. The big idea is that completing the square isn't a random trick, it's the one procedure that generates vertex form AND the quadratic formula, so understanding it once explains both.
Given a quadratic in standard form with a leading coefficient of 1, complete the square to rewrite it in vertex form.
Use the discriminant b^2-4ac to predict whether a quadratic has two real roots, one repeated real root, or two complex roots, before solving the equation.
Explain why completing the square on ax^2+bx+c derives the quadratic formula, connecting each step of the general derivation to the corresponding step in a numeric example.
Given a real-world scenario modeled by a quadratic (projectile height, rectangular area, revenue), decide which algebraic form of the quadratic (standard, factored, or vertex) to write before solving, and justify the choice by naming the graph feature the question asks for.
Simplify arithmetic expressions involving i, including powers of i and multiplication of complex numbers in a+bi form, using i^2=-1.
Explain why the complex numbers were invented as a closure of the real numbers under square roots of negative numbers, comparing this invention to the earlier extension of the number system by negative numbers or irrational numbers.
Given an unfamiliar equation that is quadratic in form but disguised (e.g., a quartic reducible by substitution, or a quadratic with irrational or literal coefficients not resembling any worked example), determine an appropriate solution method and solve it.
Given a graph of a parabola with no visible equation, generate all three algebraic forms (standard, factored where real roots exist, vertex) consistent with the graph, and identify which piece of information the graph does not uniquely determine (the leading coefficient's exact value versus its sign) without an additional point.
Recall the definition of i and state that i^2 = -1.
Classify a set of given quadratic expressions by which form (standard, factored, vertex) they are already written in, and identify what information is immediately visible in each.
Same factoring skills from Unit 1, now applied to polynomials with more than two factors, cubics, quartics, and beyond. Your child learns that a factored equation and a graph's x-intercepts are the same information in two different outfits, that a repeated factor changes the shape of the graph at that point, and that you can predict which way the far left and right ends of a graph point without plotting a single value.
Given a factored polynomial, identify each zero and state its multiplicity by counting repeated linear factors.
Divide a polynomial by a linear binomial using long division and state the quotient and remainder.
Explain why the Remainder Theorem guarantees that p(a) = 0 if and only if (x - a) is a factor, connecting the division algorithm to the factor-zero relationship.
Use the Rational Root Theorem to generate a finite list of candidate zeros for a degree 3-4 polynomial with integer coefficients, and test candidates using synthetic division.
Factor a degree-3 or degree-4 polynomial by grouping when four or more terms share a groupable structure.
Predict whether a zero's graph crosses or touches (flattens against) the x-axis based on whether its multiplicity is odd or even.
Determine end behavior of a polynomial from the parity of its degree and the sign of its leading coefficient, without evaluating any points.
Sketch a rough graph of a polynomial given its factored form, using zeros, multiplicities, and end behavior together, without plotting intermediate points.
Construct a polynomial's equation in factored form given a verbal description of its zeros with multiplicities, its end behavior, and one additional point to fix the leading coefficient.
Verify a constructed polynomial equation by expanding it and checking that the given point satisfies the expanded form.
Compare a polynomial's factored form and its expanded (standard) form and identify what each form makes visible or hides (zeros vs. sum/leading behavior).
This unit is about fractions where the top and bottom are polynomials. The core idea is that these functions behave normally almost everywhere, except at a few specific breakdown points, and whether a breakdown point is a harmless hole or an actual asymptote (a line the graph can't touch) is something you can tell from the factored equation alone, without even seeing the graph.
Given a rational expression in factored or factorable form, state its domain by identifying every value that makes the ORIGINAL (unsimplified) denominator equal to zero.
Classify a given break in a rational function's graph as a hole or a vertical asymptote by determining whether the corresponding factor cancels between numerator and denominator, and explain why cancellation produces a removable point rather than unbounded behavior.
Multiply and divide two rational expressions with polynomial numerators/denominators by factoring first, then canceling common factors before multiplying out.
Add and subtract rational expressions with unlike polynomial denominators by constructing a correct least common denominator built from the factored form of each denominator.
Given two rational expressions that appear different, determine whether they represent the same function everywhere or differ at specific excluded points, by comparing their domains after full factoring.
Solve a rational equation by clearing denominators using the LCD, then determine which resulting solutions are valid by checking each one against the original equation's restrictions.
Predict the horizontal asymptote (or absence of one) for a rational function by comparing the degrees of its numerator and denominator, extending end-behavior reasoning about dominant terms from polynomial functions to ratios of polynomials.
Given two visually similar graphs of rational functions, one with a hole and one with a vertical asymptote at the same apparent location, justify the structural difference using only the factored equations, without relying on visual features of the graph.
Construct a rational function equation that satisfies a specified combination of features (e.g., exactly one hole, one vertical asymptote, and a given horizontal asymptote), working backward from graphical structure to symbolic form.
Two things happen here that connect at the end. First, exponent rules your child already trusts (like x^2 * x^3 = x^5) get extended to fractional exponents, so square roots and cube roots turn out to be the same thing as exponents in disguise. Second, 'inverse' gets a real definition, not every function can be undone, and the horizontal line test tells you which ones can.
Given the integer exponent product rule, derive the value that a rational exponent like x^(1/2) or x^(1/3) must represent, and state the general rule x^(m/n) = (n-th root of x)^m.
Simplify radical and rational-exponent expressions (including products, quotients, and powers of powers) by applying integer exponent rules, and rationalize a denominator containing a single radical term.
Solve a radical equation by isolating the radical, raising both sides to the appropriate power, solving the resulting equation, and checking every candidate solution in the ORIGINAL equation to identify extraneous solutions.
Explain why squaring both sides of a radical equation can introduce a solution that does not satisfy the original equation, in terms of which values of x make the squared equation true versus the original equation true.
Given a function's graph or equation, determine whether it is one-to-one using the horizontal line test (from a graph) or by reasoning about repeated outputs (from an equation), and state whether an inverse function exists.
Compare a one-to-one function's graph against a many-to-one function's graph (contrasting cases) to identify what structural feature of the many-to-one graph makes the swap-coordinates procedure fail to produce a function, and generalize this into the horizontal line test rule.
Find the algebraic inverse of a one-to-one function by swapping x and y and solving for y, and verify the result is correct by showing f(g(x)) = x and g(f(x)) = x.
Given a squaring-type function that fails the horizontal line test, restrict its domain to the smallest piece that both contains a given reference point and makes the function one-to-one, then find and verify the inverse of that restricted function.
Determine, without graphing, whether an inverse function exists for a given algebraic rule from an unfamiliar function family, and justify the determination using only algebraic reasoning about repeated outputs.
Recall the meaning of the discriminant (from Unit 1) in a new context and identify an extraneous solution in a rational equation (from Unit 3) on an un-cued delayed check.
This unit explains why money in a savings account, a shrinking population of bacteria, and radioactive decay all follow the same kind of math, and why a logarithm isn't some separate weird topic, but just the question 'what power do I need?' asked formally. The single most important skill in this unit is telling from a table of numbers whether something is growing by the same AMOUNT each time (linear) or the same PERCENT each time (exponential), mixing these up is the most common real-world math mistake there is.
Given a table of values, determine whether it shows constant absolute change, accelerating change, or constant relative (percent) change, and name the family (linear, quadratic, exponential).
Write an exponential function of the form y = a(1+r)^t or y = a(1-r)^t directly from a verbal description of a percent increase or decrease situation.
Explain why log_b(x) is defined as the inverse of b^x, using the inverse-function schema (swap x and y, undo the operation) established in Unit 4.
Derive the product, quotient, and power properties of logarithms from the corresponding exponent rules rather than recalling them as separate rules.
Solve exponential equations of the form ab^(ct) = d by rewriting as a logarithm and evaluating with technology.
Solve logarithmic equations, including checking for extraneous solutions created by the restricted domain of the logarithm.
Given a real-world scenario (investment, population, decay, or a table framed as a mystery context) not previously seen in class, select the correct function family and construct a model that fits, defending the choice using the appropriate test.
Compare a constant-percent-change process to a constant-absolute-change process that produce similar values over a short early interval, and explain why they diverge over a longer time frame.
Sine and cosine get rebuilt from the ground up using the unit circle, instead of leftover right-triangle ratios from geometry. The new ideas are radians (an angle measure connected to actual arc length, not an arbitrary unit), sine and cosine as coordinates that repeat forever, and treating things like daylight hours or a Ferris wheel as transformation problems, using the same shift/stretch/reflection vocabulary from Units 1, 2, and 5.
Given an angle measured as a rotation, state the radian measure as the length of the corresponding arc on a unit circle, and convert between degree and radian measure.
Given any angle in standard position (including angles beyond 360 degrees or negative angles), identify the coordinates of the corresponding point on the unit circle as (cos theta, sin theta) and state the sign of each in each quadrant.
Explain why sine and cosine are periodic functions with period 2pi by reasoning from repeated full rotations around the unit circle, and contrast this explicitly with the monotonic long-run behavior of exponential functions from Unit 5.
Given a sine or cosine function in the form y = A sin(B(x - C)) + D, identify amplitude, midline, period, and horizontal shift, and graph the function by applying vertical stretch, vertical shift, and horizontal stretch to the parent function.
Given a real periodic data set (e.g., daylight hours by day of year) presented as a table or scatterplot, determine amplitude, midline, and period directly from the data features and write a sine or cosine equation that models it.
Use a constructed sine/cosine model to predict a value at an input outside the range of the original data set, and justify why the periodic structure supports extrapolation differently than a polynomial or exponential model would.
Prove that sin^2(theta) + cos^2(theta) = 1 follows from the Pythagorean theorem applied to the unit circle, and use the identity to find one trig value given another and the quadrant.
Given a novel periodic phenomenon described only in words and numbers (not sine/cosine language), such as a Ferris wheel or a pendulum, decide whether a sine/cosine model applies and construct one, generalizing the daylight-hours procedure to a structurally different but analogous situation without being told which family fits.
No new kind of growth here, this unit shows that arithmetic sequences are just linear functions and geometric sequences are just exponential functions, both restricted to whole-number inputs (1st term, 2nd term, 3rd term...), wearing new vocabulary. The unit closes with adding up a finite list of geometric terms (genuinely new), and a first honest look at adding infinitely many numbers and getting a finite answer.
Given a sequence presented as a list of terms, a table, or a context, identify whether it is generated by repeated addition or repeated multiplication and state which.
Given the first term and common difference (or ratio) of a sequence, generate the explicit formula a(n) = a1 + (n-1)d [or a(n) = a1 * r^(n-1)] and use it to compute a specified term.
Explain why the explicit formula for an arithmetic sequence has the same algebraic structure as a linear function's slope-intercept form, and why the geometric explicit formula mirrors exponential y = a*b^x, by identifying which part of each formula plays the role of rate/ratio and which plays the role of starting value.
Given a recursive formula for a sequence, produce the equivalent explicit formula, and given an explicit formula, produce the equivalent recursive formula.
Derive the formula for the sum of a finite geometric series by examining the structure of S - rS for a specific numeric series, then generalize the pattern to symbolic a, r, n.
Given a real-world context (loan payments, medication decay, bouncing ball height, population with constant migration), determine whether an arithmetic model, a geometric model, or neither fits, and justify the choice using the pattern of successive differences or ratios in the data.
Given an infinite geometric series with |r| < 1 presented visually (e.g., a square repeatedly divided in half) and numerically, explain informally why the partial sums approach a finite limit and compute that limit using S = a/(1-r).
Given two unfamiliar sequences side by side, one arithmetic and one geometric, with surface features deliberately similar (same first term, similarly sized second term), select the correct classification and formula type for each and explain what distinguishes them, without being told in advance which is which.
This is the last unit, and it's where all seven prior function families get put to work on real, messy data. The first half is about picking the right kind of curve to fit data and checking how well it actually fits (using leftover errors called residuals). The second half switches gears entirely, from fitting models to trusting them, sampling bias, correlation versus causation, and reading conditional probability from a table, so your child can push back on a sloppy headline instead of just believing it.
Given a data set's mean and standard deviation, estimate the percentage of values within 1, 2, or 3 standard deviations using the empirical rule.
Judge, from a histogram or dot plot, whether a normal-distribution model is an appropriate fit for a given data set, citing skew, bimodality, or outliers as disqualifying features.
Classify a scatter plot's pattern of first differences, ratios, or curvature to select linear, quadratic, or exponential as the candidate regression family, applying Unit 5's constant-difference/constant-ratio test.
Compute residuals for a fitted regression model and interpret a residual plot's pattern as evidence for or against the chosen function family.
Given an unfamiliar real data set with no family cue in the prompt, select, fit, and justify a function family using residual evidence, then state an explicit limitation on extrapolation.
Distinguish a random sample from a convenience or self-selected sample in a described study, and identify how that sampling method could bias reported results.
Explain why a strong correlation between two variables in a study does not by itself establish that one causes the other, using a specific plausible confound.
Calculate a conditional probability from a two-way frequency table and use it to evaluate whether a reported study result depends on an unstated condition.
Critique a real published or simulated news report of a study by identifying at least two of: sampling flaw, correlation/causation overreach, margin-of-error omission, or extrapolation beyond the data range.
Compare two data sets with visually similar scatter plots but different residual patterns to determine which one exponential and which one is quadratic, using structural features rather than surface curve shape alone.
From the parent guide
This is the algebra course that takes "graphing parabolas" from Algebra I and blows it out into six more families of functions, polynomials, rational expressions, radicals, exponentials/logs, trig, and finally sequences and statistics. The thread running through all eight units is one question asked over and over in new clothes: given an equation, what does the graph look like, and given a graph or a real situation, what's the equation? Your child will get a lot of practice moving between an equation, a graph, a table, and a word problem, for each of these families, and by the end they should be able to look at data or a situation and say "that's exponential, not linear" or "that needs a hole, not an asymptote" and explain why. It's a full year of building one increasingly large toolbox rather than eight unrelated topics.
Unit 1 · what to expect
This unit takes quadratics, which your child can already factor and graph from Algebra I, and gives them three ways to write the same equation (standard, factored, vertex), a way to predict how many solutions an equation has before solving it (the discriminant), and a brand-new kind of number for when the answer to a square root is negative. The big idea is that completing the square isn't a random trick, it's the one procedure that generates vertex form AND the quadratic formula, so understanding it once explains both.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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