When a = - there are three points of intersection and when a = there is one point of intersection. The next set of values for a where the function intersects the circle in points only occurs when < a <. They should have seen already that when two segments were tangent to the circle the system had two solutions and they already determined the value of a at this tangency to be a =. A: Students should move point A and observe the values of parameter a (the y coordinate of point A) in the Variables toolbox or on the graph. 3Ĥ Q5: What are the values of parameter a when the system has one two three or four solutions. Q4: How many solutions can this system have when parameter a is changing? A: Expected answer: the system can have one two three four or no solutions. The main task of this part of the problem is not to solve the system of equations but to consider the problem qualitatively.

Part 3 Investigation of the Number of Solutions In this part of the problem students investigate conditions when the system of equations has one two three or four solutions. After students solve this problem delete the measurement from the screen. a = for a and decide what values of a will lead to no solutions 4. The distance value will appear on the screen. Choose Distance/Length from the Calculate toolbox Symbolic tab. Select the center of the circle and one of the segments (hold the CTRL key to select two objects simultaneously). In order to do that the distance from the center of the circle to the segment should be equal to the radius of the circle.ģ The following steps should be followed in order to do that. Can you find these values? A: In order to answer this question students should realize that they need to determine values of parameter a when the line segments are tangent to the circle. Q: Are there any other values of parameter a when the system has no solutions? A: It is expected that students will move point A down until the graph of the function does not intersect the circle and their answer will be: yes there are values of a when there are no solutions. Students can see that when a > the function no longer intersects the circle so there are no solutions. A: It is expected that students will be moving the point A up and observing the values of parameter a in the Variables toolbox. Check that the system of equations has no solutions. Part When the System Has No Solutions Q: Consider a >. The problem is now set up and ready for investigation. The drawing is shown on the picture below. Select point O click Coordinate in the Constrain toolbox and type (0 a). In the edit box enter to fix the radius of the circle to be constant and equal to. Select the circle and choose Radius from the Constrain toolbox. Return to select mode and double-click the label of the center of the circle and re-name it to be point O. Select Circle from the Draw toolbox and construct a circle with the center at the origin. If the construction is made correctly the shape of the graph should remain the same and the point A should remain on the y-axis. As a result the graph should look like the function y = x on the restricted domain shown below.Ģ 5. Repeat steps for the other segment AC with the slope. In the data entry box type for the slope followed by the Enter key. Select the line segment and then choose Slope from the Constrain toolbox. Click the selection arrow to return to select mode. Select Line Segment from the Draw toolbox and construct segment AB with point A at an arbitrary position on the y-axis. If axes do not appear in the blank document create them by clicking the Toggle grid and axes icon on the top toolbar. Problem: Investigate the following system of equations with a parameter and find its solutions: y = x + a x + y = Part Setting Up the Problem in Geometry Expressions. Students should know that the equation y = f(x) + a corresponds to the translation of the graph of the function y = f(x) by a along the y-axis. Students should know graphs of linear functions and the equation of a circle with the center at the origin. Prior Knowledge Students should know the graphical method of solving equations and inequalities without parameters. Students obtain solutions of systems of equations (inequalities) in symbolic form.

Students apply geometric transformations of graphs of functions in order to solve systems of equations (inequalities) with parameters. Students apply a graphical method for solving systems of equations (inequalities) with parameters. 1 Solving Systems of Equations (Inequalities) with Parameters Time required 90 5 minutes Teaching Goals.

Inequalities system of equations solver