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.

original figure	image

In this section, students focus on proving geometric theorems algebraically. First, students connect the point-slope form of a line to their work with translations of lines. They then build on this work to prove that parallel lines have the same slope. Students then prove that a quadrilateral with opposite sides that lie on two pairs of parallel lines must be a parallelogram.

Next, students prove that the slopes of perpendicular lines are opposite reciprocals, or that the product of the slopes must be -1. Then students apply theorems about parallel and perpendicular lines to show that two noncollinear lines that are perpendicular to the same line are parallel to each other.

Students then inspect curves in the coordinate plane, as they examine the intersection points of circles and lines. They use the Pythagorean Theorem to find points that lie on both the line and the circle. Then they apply theorems about slope and the Pythagorean Theorem to categorize quadrilaterals in the plane. 

Next, students examine ratios, as they partition segments using weighted averages. They also apply this method to determine a coordinate definition of dilation.

Finally, students have an opportunity to apply work with slopes and the Pythagorean Theorem as they explore additional triangle theorems.

This section focuses on the features of equations and graphs of circles and parabolas.

In previous units, students defined a circle as the set of points equidistant from a given center. They connect this definition to their work with the Pythagorean Theorem in order to generalize an equation of a circle with a radius and a center as . 

Next, students revisit the distributive property as they write a squared binomial as a perfect square trinomial and vice versa.  

Then, students determine constants needed to complete the square for trinomial expressions and rewrite an equation for a circle in the form in order to determine its center and radius.

Next, students examine the graphical properties of a parabola, including the focus and directrix. They observe that any point is the same distance from the focus and the directrix, and that this property can be used to define a parabola. 

Finally, students use the Pythagorean Theorem to calculate the distance between a point and the focus, and compare it to the vertical distance between that point and the directrix. Students use the fact that these distances must be equal when they write a general equation for a parabola where the directrix is a horizontal line.