Modeling with Quadratic Functions
Lesson Plans and Worksheets for Algebra I
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More Lessons for Algebra I
Common Core For Algebra I
Examples, solutions, and videos to help Algebra I students learn how to write the quadratic function described verbally in a given context. They graph, interpret, analyze, check results, draw conclusions, and apply key features of a quadratic function to real-life applications in business and physics.
New York State Common Core Math Algebra I, Module 4, Lesson 23
Algebra I, Module 4, Lesson 23 Worksheets
Mathematical Modeling Exercise 1
Chris stands on the edge of a building at a height of 60 ft. and throws a ball upward with an initial velocity of 68 ft. per second. The ball eventually falls all the way to the ground. What is the maximum height reached by the ball? After how many seconds will the ball reach its maximum height? How long will it take the ball to reach the ground?
a. What units will we be using to solve this problem?
b. What information from the contextual description do we need to use in the function equation?
c. What is the maximum point reached by the ball? After how many seconds will it reach that height? Show your reasoning.
d. How long will it take the ball to land on the ground after being thrown? Show your work.
e. Graph the function of the height (h) of the ball in feet to the time (t) in seconds. Include and label key features of the graph, such as the vertex, axis of symmetry, and x- and y-intercepts.
Lesson 23 Summary
We can write quadratic functions described verbally in a given context. We also graph, interpret, analyze, or apply key features of quadratic functions to draw conclusions that help us answer questions taken from the problem’s context.
- We find quadratic functions commonly applied in physics and business.
- We can substitute known x- and y-values into a quadratic function to create a linear system that, when solved, can identify the parameters of the quadratic equation representing the function.
Lesson 23 Problem Set Sample Solutions
1. Dave throws a ball upward with an initial velocity of 23 ft. per second. The ball initially leaves his hand 5 ft. above the ground and eventually falls back to the ground. In parts (a)–(d), you will answer the following questions: What is the maximum height reached by the ball? After how many seconds will the ball reach its maximum height? How long will it take the ball to reach the ground?
a. What units will we be using to solve this problem?
b. What information from the contextual description do we need to use to write the formula for the function of the height of the ball versus time? Write the formula h for height of the ball in feet, h(t), where t stands for seconds.
c. What is the maximum point reached by the ball? After how many seconds will it reach that height? Show your reasoning.
d. How long will it take for the ball to land on the ground after being thrown? Show your work.
e. Graph the function of the height of the ball in feet to the time in seconds. Include and label key features of the graph such as the vertex, axis of symmetry, and x- and y-intercepts.
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