Grade 11

Determine slopes from points and graphs, recognizing positive, negative, zero, and undefined slopes, and write equations of lines in slope-intercept, point-slope, or standard form, including horizontal and vertical lines and parallel or perpendicular relationships. Use the distance and midpoint formulas for points in the two-dimensional Cartesian plane, and determine or interpret circle equations in standard form using their centres and positive radii. Solve coordinate geometry problems involving lines and circles by applying slope, distance, midpoint, substitution, and elimination to find unknown coordinates, intersections, parallel or perpendicular relationships, tangency, and zero, one, or two intersection points.

Classify data as categorical or quantitative and as discrete or continuous, evaluate sampling methods for undercoverage, convenience or voluntary response, and nonresponse bias, and recognize that random sampling reduces selection bias without eliminating sampling variability. Calculate and interpret the mean, median, mode, range, interquartile range, variance, and standard deviation for lists or frequency tables, and construct and interpret histograms and box plots using distribution, center, spread, skewness, and possible outliers. Construct and interpret scatter plots, describe the direction and strength of linear or nonlinear association while distinguishing correlation from causation, and use lines of best fit to make and evaluate interpolated or extrapolated predictions.

Students distinguish linear additive change from exponential multiplicative change by using differences, ratios, tables, equations, graphs, and comparisons of initial values and long-term behavior. They determine initial values and growth or decay factors, evaluate functions of the form $f(x)=ab^x$, recognize growth and decay, write models from contexts and equally spaced tables, and interpret basic graphs, transformations, intercepts, domain, range, end behavior, and horizontal asymptotes. They solve simple exponential equations by rewriting both sides with a common positive base and equating exponents, without using logarithmic methods, regression, continuous compounding, or generalized exponential models.

Students calculate simple and compound interest, accumulated amounts, future values, and present values using consistent rates, compounding frequencies, and time periods, and compare nominal and effective annual rates. They model regular deposits as annuities and calculate fixed-rate loan payments, amortization, total repayment, and total interest, distinguishing interest from principal as balances change. They compare investment and financing options using formulas, tables, graphs, fees, terms, payment schedules, inflation effects, and total costs within standard fixed-rate models.

Students distinguish relations from functions using mappings, tables, equations, graphs, and the vertical line test, and evaluate standard algebraic functions using function notation while respecting domain restrictions. They determine domains from equations, ranges from graphs, zeros, positive and negative intervals, and increasing, decreasing, or constant intervals, accounting for endpoints, holes, turning points, zeros, and undefined inputs. They interpret domain, range, intercepts, extrema, symmetry, discontinuities, end behavior, and other key graph features in mathematical and real-world contexts, including appropriate units and restrictions such as nonnegative time or whole-number quantities.

Students convert between exponential and logarithmic forms, interpret logarithms as exponents, and evaluate logarithms with exact integer values, including values involving 1 and negative integer exponents. Students apply the product, quotient, and power laws to expand and condense real logarithmic expressions with positive arguments, using familiar integer and rational powers while recognizing that logarithms do not distribute over sums or differences. Students solve simple logarithmic equations and exponential equations by using inverse relationships and logarithm properties, checking that bases are positive and not 1, all arguments are positive, and solutions satisfy the original equations.

Students add, subtract, multiply, and evaluate one-variable polynomial functions with numerical coefficients, classify them by degree from 0 through 5, and identify their standard-form leading coefficients. They factor polynomial expressions completely over the specified number system and use factored form and the zero-product property to solve polynomial equations, determine real zeros and x-intercepts, and recognize repeated zeros and their multiplicities. Students predict end behavior from degree and leading coefficient and sketch polynomial graphs by combining end behavior with zeros and multiplicity-based crossing or touching behavior.

Students simplify rational expressions by factoring polynomials, preserving denominator restrictions, and multiplying or dividing by canceling common factors rather than individual terms. They add and subtract rational expressions by finding a least common denominator, combining numerators correctly, and working with linear or common quadratic factors. They determine valid domains, solve simple rational equations by clearing denominators, reject extraneous solutions through checks in the original equation, and interpret rational expressions as rates, proportions, or relationships in context.

Recognize arithmetic and geometric sequences from lists, tables, graphs, recursive rules, or explicit forms, distinguishing constant additive differences from constant multiplicative ratios. Determine common differences and ratios, find terms, and write explicit formulas using $a_n=a_1+(n-1)d$ and $a_n=a_1r^{n-1}$, including positive, negative, fractional, zero, and alternating patterns where applicable. Calculate finite arithmetic sums with $S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}[2a_1+(n-1)d]$ and finite geometric sums with $S_n=\frac{a(1-r^n)}{1-r}$ for $r\ne1$ or $S_n=na$ for $r=1$, excluding infinite-series convergence and complex or advanced generalized cases.

Solve one-variable linear and compound inequalities using equivalent operations, reversing the inequality when multiplying or dividing by a negative, and represent solutions with number lines, intervals, or set notation. Solve two-variable systems of linear equations algebraically and graphically, including systems with unique, no, or infinitely many solutions, and solve linear–quadratic systems by substitution or graphing to identify zero, one, or two real intersection points. Graph linear inequalities and systems as half-planes with correctly included or excluded boundaries, interpret their intersections as feasible regions and solution sets, and use ordered pairs, units, and relevant corner points to assess feasible choices.

Students identify horizontal and vertical translations, stretches and compressions, and reflections across the $x$- and $y$-axes, interpreting their effects on equations, graphs, domains, ranges, intercepts, and key points. Students graph and describe combined transformations of familiar parent functions, including linear, quadratic, absolute-value, square-root, and reciprocal functions, using $g(x)=a f(b(x-h))+k$ and the mapping $(x,y)\mapsto\left(\frac{x}{b}+h,ay+k\right)$, with attention to the reciprocal effect of horizontal factors. Students determine transformed equations from graphs by inferring parameters from vertices, intercepts, endpoints, asymptotes, and corresponding points, and connect equations, mappings, tables, and graphical features.

Convert between degree and radian measures using $180^\circ=\pi$, identify coterminal and reference angles in standard position, and use the unit circle, quadrant signs, and special angles $0^\circ,30^\circ,45^\circ,60^\circ,$ and $90^\circ$ to determine exact sine, cosine, and tangent values. Graph and interpret transformed sine and cosine functions, determining amplitude, period, phase shift, vertical displacement, midline, range, key points, zeros, maxima, minima, reflections, and periodic extensions from equations in degrees or radians. Model real-world periodic phenomena with $y=A\sin(B(x-C))+D$ or $y=A\cos(B(x-C))+D$, relating parameters to measurable amplitude, period, phase shift, midline, units, and values while distinguishing periodic behavior from linear change.