Note that I made a mistake in the graph for #6. The multiplicity 2 is at x = 0 — so the graph should have “bounced” at x = 0 NOT at x = -2 like I did on my sketch 🤦🏻♀️ that graph should cross through at x = -2 then go up towards x = 0 and “bounce” back down off of x = 0.
This product is based on the IM K-12 MathTM authored by Illustrative Mathematics® and offered under a CC BY 4.0 License.
Unit Title: Polynomials
Title: Finding Intersections
Common Core Standards in This Lesson:
CCSS.HSA-REI.C.7
CCSS.HSA-REI.C.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line 𝘺 = –3𝘹 and the circle 𝘹² + 𝘺² = 3.
CCSS.HSA-REI.D.11
CCSS.HSA-REI.D.11: Explain why the 𝘹-coordinates of the points where the graphs of the equations 𝘺 = 𝘧(𝘹) and 𝘺 = 𝑔(𝘹) intersect are the solutions of the equation 𝘧(𝘹) = 𝑔(𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧(𝘹) and/or 𝑔(𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
CCSS.MP7
CCSS.MP7: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝑥² + 9𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝑥 – 𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝑥 and 𝑦.
