Graphing Systems of Equations

Graphing Systems of Equations

Student Outcomes

• Students develop facility with graphical interpretations of systems of equations and the meaning of their solutions on those graphs. For example, they can use the distance formula to find the distance between the centers of two circles and thereby determine whether the circles intersect in 0, 1, or 2 points.
• By completing the squares, students can convert the equation of a circle in general form to the center-radius form and, thus, find the radius and center. They can also convert the center-radius form to the general form by removing parentheses and combining like terms.
• Students understand how to solve and graph a system consisting of two quadratic equations in two variables.

New York State Common Core Math Algebra II, Module 1, Lesson 32

Worksheets for Algebra II, Module 1, Lesson 32

Classwork

Opening Exercise

Given the line 𝑦 = 2𝑥, is there a point on the line at a distance 3 from (1, 3)? Explain how you know.

Draw a graph showing where the point is

Exercise 1

Solve the system (𝑥 − 1)2 + (𝑦 − 2)2 = 22 and 𝑦 = 2𝑥 + 2.
What are the coordinates of the center of the circle?
What can you say about the distance from the intersection points to the center of the circle?
Using your graphing tool, graph the line and the circle.

Example 1

Rewrite 𝑥2 + 𝑦2 − 4𝑥 + 2𝑦 = −1 by completing the square in both 𝑥 and 𝑦. Describe the circle represented by this equation.

Using your graphing tool, graph the circle.

In contrast, consider the following equation: 𝑥2 + 𝑦2 − 2𝑥 − 8𝑦 = −19
What happens when you use your graphing tool with this equation?

Exercise 2

Consider a circle with radius 5 and another circle with radius 3. Let 𝑑 represent the distance between the two centers.
We want to know how many intersections there are of these two circles for different values of 𝑑. Draw figures for each case.
a. What happens if 𝑑 = 8?
b. What happens if 𝑑 = 10?
c. What happens if 𝑑 = 1?
d. What happens if 𝑑 = 2?
e. For which values of 𝑑 do the circles intersect in exactly one point? Generalize this result to circles of any radius.
f. For which values of 𝑑 do the circles intersect in two points? Generalize this result to circles of any radius.
g. For which values of 𝑑 do the circles not intersect? Generalize this result to circles of any radius.

Example 2

Find the distance between the centers of the two circles with equations below, and use that distance to determine in how many points these circles intersect.

Exercise 3

Use the distance formula to show algebraically and graphically that the following two circles do not intersect.

Example 3

Point 𝐴(3, 2) is on a circle whose center is 𝐶(−2, 3). What is the radius of the circle?
What is the equation of the circle? Graph it.

Use the fact that the tangent at 𝐴(3, 2) is perpendicular to the radius at that point to find the equation of the tangent line. Then graph it.
Find the coordinates of point 𝐵, the second intersection of the 𝐴𝐶 and the circle.
What is the equation of the tangent to the circle at (−7, 4)? Graph it as a check
The lines 𝑦 = 5𝑥 + 𝑏 are parallel to the tangent lines to the circle at points 𝐴 and 𝐵. How is the 𝑦-intercept 𝑏 for these lines related to the number of times each line intersects the circle

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