Unit 4, Lesson 19
Real and Non-Real Solutions (Optional)
Algebra 2
19.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that although it may be difficult to tell from a function's expression whether it will equal 0 for any real values of , it is relatively straightforward to tell this from looking at the function's graph. This will be useful when students examine the number and type of solutions for a quadratic equation in a later activity. While students may notice and wonder many things about these quadratic representations, the connection between the number of real solutions and the number of intersections on the graph are the important discussion points.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the effect on the graph of changing the constant term, and the effect this has on the number of real -values at which the function is 0. Students will explore the effect of the other terms in later activities.
Launch
Arrange students in groups of 2. Display the graphs and functions for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
when .
when .
when .
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the functions and graphs. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the connection between the number of real solutions and the number of times the graph intersects the -axis does not come up during the conversation, ask students to discuss this idea.
Student Lesson Summary
By looking at the graph of a quadratic function, we can tell whether it has real zeros or not. The real zeros are the -coordinates of the points where the graph touches the -axis. So, if the graph doesn't touch the -axis at all, then the function does not have real zeros. If that happens, we won't be able to tell exactly what the zeros are just by looking at the graph, but we will be able to find them by setting the function equal to 0 and solving algebraically.
We can also tell whether has real zeros by using the quadratic formula strategically, since the quadratic formula tells us the -values that make a quadratic function be 0. For reference, here is the quadratic formula:
Imaginary numbers are the square roots of negative numbers, so it will be helpful to look at the part of the quadratic formula that involves a square root, which is . If is greater than , then the number under the radical will be negative, which means that the zeros of the quadratic function are not real.
For example, consider the quadratic function . How can we tell if it has real zeros?
One way is to look at the graph, shown here. Since its graph doesn't intersect the -axis, this function doesn't have any real zeros.
We can also tell by checking whether is greater than . Since , , and , , so the zeros of this function have an imaginary part. We can find out exactly what the zeros are by using the quadratic formula or completing the square.