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Interpret real-world situations as functions in context. Describe the relationships qualitatively, paying attention to the general shape of the graph without concern for specific numerical values. Describe the relationships qualitatively to determine where the function is increasing or decreasing, linear or nonlinear, or both. Proceed from left to right and describe what happens to the output as the input value increases. From the graph alone, infer meaning from a function.

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Interpret real-world situations as functions in context. The slope of a vertical line is undefined and the slope of a horizontal line is 0. Either of these cases might be considered “no slope.” Thus, the phrase “no slope” should be avoided because it is ambiguous, and “non-existent slope” and “slope of 0” should be distinguished from each other.

The starting point for a function is the first value of the input. Determine the y-intercept of a function. The rate of change is the slope of the interpreted function of the graph. Construct functions to model real-world situations. Construct a function to model a linear relationship between two quantities.

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Two-way tables organize data based on two categorical variables. Construct two-way tables summarizing data on two categorical variables collected from the same subjects. Analyze frequencies shown in two-way tables of categorical data to look for the association. Describe the association of categorical variables in the content of the problem.

Add or subtract percentage values from two-way tables to see if corresponding associations are consistent. Compute marginal sums or marginal percentages.

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Graph bivariate measurement data from an equation that associates two variables. Interpret the graph of a linear model in the context of a problem. After a line is fit through the data, the slope of the line is approximated and interpreted as a rate of change. Analyze the line of best fit in the context of the variables. Use the line of best fit to solve problems.

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Graph paired measurements that may be associated with a scatter plot. Look for a positive or negative trend in the cloud of points. Identify a nonlinear pattern. Identify a strong association (most of the points increase or decrease in correspondence with the associated values) or a weak association (some of the points increase or decrease in correspondence with the associated values of the variables).

Describe groups of points as clusters. Describe points that are far off the trending points as outliers. If the points generally trend in a linear fashion (trend in a straight line) they are said to have a linear association. When paired associated measures trend in a generally linear fashion, use a line of best fit to show the general trend. The line of best fit is an approximation to show the associative trend.

Determine if a line of best fit is placed accurately by analyzing the location of points on the scatter plot. After a line is fit through the data, the slope of the line is approximated and interpreted as a rate of change.

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Graph paired measurements that may be associated with a scatter plot. Look for a positive or negative trend in the cloud of points. Recognize a nonlinear pattern (curved pattern). Recognize a strong association (most of the points correspond by increasing or decreasing in correspondence with the associated values) or weak association (some of the points correspond by increasing or decreasing in correspondence with the associated values of the variables).

Describe groups of points as clusters. Describe points that are far off the trending points as outliers. If the points generally trend in a linear fashion (trend in a straight line) they are said to have a linear association. Interpret scatter plots for bivariate measurement data.

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A cone is formed by drawing segments from a two-dimensional figure to a point that lies outside the plane of the figure; to be precise, it is the set of all line segments from the point to the figure. The two-dimensional figure is called the base of the cone, and the point, its apex. Apply volume formulas for cones, pyramids, and spheres by substituting correct values for variables and evaluating. Determine from real-world scenarios the measurements for the base, height, and radius of figures.

The volume of a pyramid whose base has area b and whose height is h is 1/3bh. The volume of a cylinder is πr^{2}h. The volume of a sphere is 4/3 πr³.

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The Pythagorean Theorem applies to all right triangles and states that the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the square of the two other sides, or a^{2} + b^{2} = c^{2}. Substitute values for the length of legs and hypotenuse to determine the missing side. Determine if the side lengths given form a right triangle. Determine the missing side length in a right triangle. Solve real-world and mathematical problems that involve the Pythagorean Theorem.

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The Pythagorean Theorem applies to all right triangles and states that the square of the hypotenuse (longest side opposite the right angle) is equal to the sum of the square of the two other sides, or a^{2} + b^{2} = c^{2}. Substitute values for the length of legs or hypotenuse to determine the missing side.

Given a slope on a coordinate graph, determine the length of the slope by determining perpendicular leg lengths where the slope is the hypotenuse and apply the Pythagorean Theorem to determine the length of the hypotenuse. Leg length and hypotenuse ratio will be a constant proportion regardless of the length chosen for the legs.

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The Pythagorean Theorem applies to all right triangles and states that the square of the hypotenuse (longest side opposite the right angle) is equal to the sum of the square of the two other sides, or a^{2} + b^{2} = c^{2}. The converse of the Pythagorean Theorem is used to determine if a triangle is a right triangle. It uses the relationship between the lengths of the legs and hypotenuse to prove a right angle.

Use a dissection model to show the proof of the Pythagorean Theorem.

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