In this activity, students input a function where the constants, coefficients, and exponents can be adjusted by slider bars. Students can also input data points to be graphed in the same window. This activity allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Students can also practice adjusting the parameters of an equation in order to find a function representation of their data. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

- Mathematics > General

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Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

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.?

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

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

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Combine standard function types using arithmetic operations.

(+) Compose functions.

Identify 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.

Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret the parameters in a linear or exponential function in terms of a context.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.?

Summarize 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.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Informally assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation coefficient of a linear fit.