24.1: Equations of Two Lines and A Curve (10 minutes)
This warm-up activates some familiar skills for writing and solving equations, which will be useful for specific tasks throughout the lesson.
It may have been a while since students thought about writing an equation for a line passing through two points. The two questions here are intentionally quite straightforward. Monitor for students taking different approaches, such as:
- plotting the points and determining the slope and \(y\)-intercept of a line passing through the points
- computing the slope by finding the quotient of the difference between the \(y\)-coordinates and difference between the \(x\)-coordinates
- considering what operation on each \(x\)-coordinate would produce its corresponding \(y\)-coordinate
Students have not yet solved a quadratic equation like the one in the second question, but they have learned and extensively practiced the skills needed to solve it. The two main anticipated approaches are:
- reasoning algebraically, by performing the same operation to each side of the equation, applying the distributive property to expand factored expressions, combining like terms, rewriting an expression in factored form, and applying the zero product property
- graphing \(y=x+1\) and \(y=(x-2)^2-3\) and observing the \(x\)-coordinate of each point of intersection
Write an equation representing the line that passes through each pair of points.
- \((3,3)\) and \((5,5)\)
- \((0,4)\) and \((\text-4,0)\)
- Solve this equation: \(x+1=(x-2)^2-3\). Show your reasoning.
Invite students taking different approaches to share their work. Ensure that students see more than one way to think about the equation representing a line for the first question, and recognize that the second equation can be solved algebraically.
24.2: The Dive (15 minutes)
Monitor for students taking different approaches, such as:
- Graphing the function and finding and interpreting points on the graph.
- Evaluating the function at relevant values.
- Writing and solving equations whose solutions answer the questions. Some equations can be solved by rewriting it as a factored expression equal to 0, while some must be solved by completing the square or using the quadratic formula.
- Rewriting the given expression in a different form—for example, rewriting it in vertex form to find the maximum value of the function.
If any students take a graphing approach and finish quickly, challenge them to verify their solution to each problem by using a second method that does not rely on graphing.
Ask students to read the task quietly and sketch a rough graph showing the diver’s height as a function of time. When a diver jumps off a board, her height increases for a short time, and then decreases until the time at which she enters the water, or the time at which her height above the water is 0 meters.
Display one of these sketches for all to see, and ask what information might be added to the graph that we know so far. Generally, we want to make sure students understand that the values along the horizontal axis correspond to the number of seconds after the jump, and values along the vertical axis correspond to the height in meters of the diver above the water.
Design Principle(s): Maximize meta-awareness; Support sense-making
Supports accessibility for: Conceptual processing; Language
The function \(h\), defined by \(h(t)=\text-5t^2+10t+7.5\), models the height of a diver above the water (in meters), \(t\) seconds after the diver leaves the board. For each question, explain how you know.
- How high above the water is the diving board?
- When does the diver hit the water?
- At what point during her descent toward the water is the diver at the same height as the diving board?
- When does the diver reach the maximum height of the dive?
- What is the maximum height the diver reaches during the dive?
Are you ready for more?
Another diver jumps off a platform, rather than a springboard. The platform is also 7.5 meters above the water, but this diver hits the water after about 1.5 seconds.
Write an equation that would approximately model her height over the water, \(h\), in meters, \(t\) seconds after she has left the platform. Include the term \(\text-5t^2\), which accounts for the effect of gravity.
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If students have trouble getting started, tell them that sometimes it is helpful to restate a question to make it about the function or its graph. For example, “When does the diver hit the water?” can be restated as “At what time is the diver 0 meters above the water?” or “What is the positive horizontal intercept of a graph representing the function?” Ask students to think about how they might restate some questions about the situation as questions about the function or its graph and write these down or share them with a partner before they get to work.
Display the graph for all to see and ask students to interpret the graph: which numbers answer which questions? Once this is settled, invite students to share other approaches to solving the problems besides interpreting coordinates on the graph.
24.3: A Linear Function and A Quadratic Function (10 minutes)
The purpose of this activity is for students to use what they know to solve an unfamiliar problem. Since it is unfamiliar, students need to make sense of the problem and demonstrate perseverence (MP1). This is a preview of solving a system consisting of a linear equation and a quadratic equation algebraically and graphically, which students will study more in depth in a future course.
Here are graphs of a linear function and a quadratic function. The quadratic function is defined by the expression \((x-4)^2-5\).
Find the coordinates of \(P, Q\), and \(R\) without using graphing technology. Show your reasoning.
Invite one or more students to demonstrate their solution, or if they got stuck, demonstrate the progress that they made.
Design Principle(s): Optimize output (for explanation)
Invite students to create a visual display of one of their solutions to one of the tasks. Then, display these around the room and provide each student with a few sticky notes. Invite them to observe one or more of their classmates’ solutions, and leave a sticky note if they have a question or observation to share. After this gallery walk, allow students time to review the feedback they recieved on their display and invite students to share anything new they learned or questions they have after seeing some of their classmates’ work.
24.4: Cool-down - Profit from A River Cruise (5 minutes)
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Student Lesson Summary
Certain real-world situations can be modeled by quadratic functions, and these functions can be represented by equations. Sometimes, all the skills we have developed are needed to make sense of these situations. When we have a mathematical model and the skills to use the model to answer questions, we are able to gain useful or interesting insights about the situation.
Suppose we have a model for the height of a launched object, \(h\), as a function of time \(t\), defined by \(h(t) = \text-4.9t^2 + 28t + 2.1\). We can answer questions such as these about the object’s flight:
- From what height was the object launched?
(An expression in standard form can help us with this question. Or, we can evaluate \(h(0)\) to find the answer.)
- At what time did it hit the ground?
(When an object hits the ground, its height is 0, so we can find the zeros using one of the methods we learned: graphing, rewriting in factored form, completing the square, or using the quadratic formula.)
- What was its maximum height, and at what time did it reach the maximum height?
(We can rewrite the expression in vertex form, but we can also use the zeros of the function or a graph to do so.)
Sometimes, relationships between quantities can be effectively communicated with graphs and expressions rather than with words. For example, these graphs represent a linear function, \(f\), and a quadratic function, \(g\), with the same variables for their input and output.
If we know the expressions that define these functions, we can use our knowledge of quadratic equations to answer questions such as:
- Will the two functions ever have the same value?
(Yes. We can see that their graphs intersect in a couple of places.)
- If so, at what input values does that happen? What are the output values they have in common?
(To find out, we can write and solve this equation: \(f(x) = g(x)\).)