Standards - Mathematics

Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. [Algebra I with Probability, 14]

Note: The graph of a relation often forms a curve (which could be a line).

Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Identify that a function $f$ is a special kind of relation defined by the equation $y = f(x)$. [Algebra I with Probability, 16]

Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Use function composition to combine different types of standard functions to write and evaluate functions. [Algebra I with Probability, 17]

Explain 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 approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. [Algebra I with Probability, 19]

Note: Include cases where $f(x)$ is linear, quadratic, exponential, or absolute value functions and $g(x)$ is constant or linear.

Graph 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, using technology where appropriate. [Algebra I with Probability, 20]

Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate. [Algebra I with Probability, 18]

Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Include linear, quadratic, exponential, absolute value, and linear piecewise. [Algebra I with Probability, 21, edited]

Distinguish between linear and non-linear functions. [Grade 8, 15a]

Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.

Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions. [Algebra I with Probability, 22]

Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f (kx)$, 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 explain the effects on the graph, using technology as appropriate. Extend from linear to quadratic, exponential, absolute value, and linear \piecewise functions. [Algebra I with Probability, 23, edited]

Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

Construct 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). [Algebra I with Probability, 25]

Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. [Algebra I with Probability, 26]

Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form $mx + b$, to exponential functions, written in the form $ab^x$. [Algebra I with Probability, 27]

For 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.

Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and general piecewise functions. [Algebra I with Probability, 28]

Calculate 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. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 29]

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Graph piecewise-defined functions, including step functions and absolute value functions.

Graph exponential functions, showing intercepts and end behavior. [Algebra I with Probability, 30]

Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 31]

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non- linear association, clustering, and outliers. [Grade 8, 18]

Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line. [Grade 8, 19]

Use a linear model of a real-world situation to solve problems and make predictions.

Describe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation. [Grade 8, 20]

Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables. [Grade 8, 21]

Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types. [Algebra I with Probability, 34]

Analyze the possible association between two categorical variables.