M.A1HS: High School Algebra I

High School Algebra I

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High School Algebra I covers linear and quadratic functions and their applications, with students writing, solving, and graphing equations and inequalities in one and two variables. Systems of equations, polynomial operations, and quadratic formula applications are central. The course closes with exponential functions, statistical modeling with scatter plots, and interpreting linear and nonlinear regression models.

M.A1HS.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.M.A1HS.10Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.M.A1HS.11Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)3 to hold, so (5^1/3)^3 must equal 5.)M.A1HS.12Rewrite expressions involving radicals and rational exponents using the properties of exponents.M.A1HS.13Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.M.A1HS.14Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.M.A1HS.15Recognize that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).M.A1HS.16Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values or find successive approximations). Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential and logarithmic functions.M.A1HS.17Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.M.A1HS.18Recognize that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).M.A1HS.19Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.M.A1HS.2Define appropriate quantities for the purpose of descriptive modeling.M.A1HS.20Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (e.g., The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.M.A1HS.21For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.M.A1HS.22Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.)M.A1HS.23Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.M.A1HS.24Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A1HS.25Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)M.A1HS.26Write a function that describes a relationship between two quantities.M.A1HS.27Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.M.A1HS.28Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.M.A1HS.29Distinguish between situations that can be modeled with linear functions and with exponential functions.M.A1HS.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.M.A1HS.30Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship or two input-output pairs (include reading these from a table).M.A1HS.31Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.M.A1HS.32Interpret the parameters in a linear or exponential function in terms of a context.M.A1HS.33Represent data with plots on the real number line (dot plots, histograms, and box plots).M.A1HS.34Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.M.A1HS.35Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).M.A1HS.36Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.M.A1HS.37Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.A1HS.38Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.M.A1HS.39Compute (using technology) and interpret the correlation coefficient of a linear fit.M.A1HS.4Interpret expressions that represent a quantity in terms of its context.M.A1HS.40Distinguish between correlation and causation.M.A1HS.41Interpret expressions that represent a quantity in terms of its context.M.A1HS.42Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).M.A1HS.43Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.M.A1HS.44Recognize that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.M.A1HS.45Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.M.A1HS.46Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.M.A1HS.47Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.M.A1HS.48Solve quadratic equations in one variable.M.A1HS.49Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.M.A1HS.5Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.M.A1HS.50Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.M.A1HS.51For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.M.A1HS.52Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.M.A1HS.53Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.M.A1HS.54Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A1HS.55Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.A1HS.56Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.M.A1HS.57Write a function that describes a relationship between two quantities.M.A1HS.58Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.M.A1HS.59Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x^3 or f(x) = (x+1)/(x-1) for x ≠ 1.M.A1HS.6Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.M.A1HS.60Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.M.A1HS.7Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.)M.A1HS.8Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)M.A1HS.9Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.M.A1HS.DSDescriptive StatisticsM.A1HS.EEExpressions and EquationsM.A1HS.LERLinear and Exponential RelationshipsM.A1HS.QFMQuadratic Functions and ModelingM.A1HS.RQRRelationships between Quantities and Reasoning with Equations
Example Problems
Solve for x:
Solve for x.
Find
Solve the system of equations using the elimination method:

Solve the following system of equations using the elimination method:

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