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Section: A1.7.7

Connect algebra to geometry

• Solve problems using systems of a linear and a quadratic equation in two variables (A-REI.C.7).
• Derive equation of a parabola given the focus and a directrix parallel to one of the axes (G-GPE.A.2).

In this section students connect equations in two variables to the geometry of the curves they define. They study systems of linear and quadratic equations. In Grade 8 they used similar triangles to explain why a line has constant slope, and thus derived the equation $y=mx+b$ for a non-vertical line. In this unit they extend this work to parabolas, explaining why the geometric definition of a parabola leads to a quadratic equation in two variables.

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1 A Linear and Quadratic System

WHAT: Students are given graphs of a linear and a quadratic function on the same axes and are asked for the coordinates of two points. Solving the task requires an interplay between geometry and algebra, going from graphs to equations and back again. Finding one of the points requires reducing the system to a quadratic equation and solving it.

WHY: The purpose of this task is to give students the opportunity to make connections between equations and the geometry of their graphs. They must read information from the graph (such as the vertical intercept of the quadratic graph or the slope of the linear one), use that information to construct and solve an equation, then interpret their solution in terms of the graph. The task also requires the basic understanding that the coordinates of the points of intersection of the graphs are the pairs of values of the variables that solve the system.

2 Defining Parabolas Geometrically

WHAT: Students are given a line and a point and asked to find points equidistant from both and to give the equation for the locus of all such points.

WHY: Although the graph of a quadratic function is often referred to as a parabola, the definition of a parabola is geometric: it is the locus of points equidistant from a point (the focus) and a line (the directrix). The purpose of this task is to show, in a particular case, that the graph of a quadratic function is a parabola in this sense.

This task can be extended to an exploration in a number of ways. The method can be adapted to any situation where the directrix is parallel to one of the axes. If geometry software is available, students can construct parabolas for any focus and directrix. Finally, they can show that the graph of any quadratic function is a parabola by first using translations to show that the graph is congruent to the graph $y=ax^2$ for some constant $a$, and then using the method of this task to show that this graph is a parabola with focus on the y-axis and directrix equal to the $x$-axis.