This section focuses on understanding transformations as a type of function. Students first recall work from earlier units as they connect rigid transformations in the coordinate plane using the Pythagorean Theorem to identify lengths of segments. Students are then introduced to function notation for transformations as they identify   as a translation,  , as a dilation with scale factor , and transformations like   as a reflection. They also use tables and figures to create their own function rules to represent transformations in the coordinate plane. Finally, students compare transformation functions and decide whether they represent rigid transformations, similarity transformations, or neither.  

Two squares on coordinate plane, origin O. Horizontal x axis from negative 2 to 6. Vertical axis from negative 4 to 4. Square A B C D with vertices A at negative 1 comma 1, B at 1 comma 1, C at 1 comma negative 1 and D at negative 1 comma negative 1. Point Q at negative point 5 comma 1. Square A prime B prime C prime D prime with vertices A prime at 4 comma 0, B prime at 6 comma 0, C prime at 6 comma negative 2, D prime at 4 comma negative 2, and Q prime at 4 point 5 comma 0.

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This section focuses on lines and their relationships to each other. Students begin by developing an equation for point-slope form of an equation: . They compare this form to other forms they have previously used, such as , , and .

This new form is especially helpful for describing a line under a translation, and students build from this representation to study parallel lines. They determine that a line that is translated gives a parallel line and show that their slopes must be identical, using both geometric and algebraic arguments. 

Next, students compare slopes of lines that are parallel. They begin with describing a 90-degree rotation about the origin using function notation: . Using this notation, they construct arguments that the product of the slopes is always -1, or that the slopes of parallel lines are opposite reciprocals.

Finally, students create equations for lines that are parallel or perpendicular to another line through a given point. 

In this section, students connect triangles and quadrilaterals in the coordinate plane to their work with equations of lines in the plane. They begin by finding the length of segments in the plane, using the Pythagorean Theorem, and then build a generalized equation to find the distance between any two points in the coordinate plane using equations like  or . 

Next, students find slopes and distances on the plane and use them to classify triangles as right or non-right, and quadrilaterals as rectangles, squares, rhombuses, or parallelograms. 

Finally, there is an optional lesson that provides an opportunity for additional practice with slopes of parallel and perpendicular lines and with creating equations for lines given a slope and a point, as students explore triangle altitudes, perpendicular bisectors, and tessellations.

While the work of this section builds toward the development of the equation of a circle, students are not expected to connect the Pythagorean Theorem with equations for circles at this time.

Segment P Q on a coordinate plane, origin O. Horizontal axis scale 0 to 50 by 10’s. Vertical axis scale 0 to 20 by 10’s. Points P(8 comma 12) and Q(53 comma 22) form segment P Q. Point N(53 comma 12) forms segments N Q and P N with dashed segments.