standard_description: 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.
problem_activity: Warm-up
The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.
Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the
best
price, however, do form a function, because there is only one possible best price for each number of bagels.
Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.
Launch
Arrange students in groups of 4. Give students a few minutes of quiet time to think about the first question, and then a couple of minutes to share their thinking with their group. Pause for a class discussion.
Invite students to share an explanation as to how each person in the situation could be right. If possible, record and display students’ reasoning for all to see. After the reasoning behind each price is shared, direct students to the table in the second question. Ask students to write down “best price” in the header of the second column and then complete the table.
Student Facing
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A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
Jada says to her friends, “Shouldn’t the total be \$13.25?”
Priya says, “I think it should be \$13.00.”
Han says, “No, I think it should be \$11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
Your teacher will give you instructions for completing the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "br": [{}], "_values": ["number", "of\u00a0bagels"]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}]}, {"td": [{"_value": "2"}, {}]}, {"td": [{"_value": "3"}, {}]}, {"td": [{"_value": "4"}, {}]}, {"td": [{"_value": "5"}, {}]}, {"td": [{"_value": "6"}, {}]}, {"td": [{"_value": "7"}, {}]}, {"td": [{"_value": "8"}, {}]}, {"td": [{"_value": "9"}, {}]}, {"td": [{"_value": "10"}, {}]}, {"td": [{"_value": "11"}, {}]}, {"td": [{"_value": "12"}, {}]}, {"td": [{"_value": "13"}, {}]}]}]}]}
Student Response
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Activity Synthesis
Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels), as well as the table showing the best price for each number of bagels bought.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["shopkeeper\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Jada\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Priya\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Han\u2019s", "price"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}, {}, {}, {}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}, {}, {}, {}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}, {}, {}, {}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}, {}, {}, {}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}, {}, {}, {}]}, {"td": [{"_value": "6"}, {"_value": "7.50"}, {"_value": "6.00"}, {}, {}]}, {"td": [{"_value": "7"}, {"_value": "8.75"}, {"_value": "7.25"}, {}, {}]}, {"td": [{"_value": "8"}, {"_value": "10.00"}, {"_value": "8.50"}, {}, {}]}, {"td": [{"_value": "9"}, {"_value": "11.25"}, {"_value": "9.75"}, {"_value": "8.00"}, {}]}, {"td": [{"_value": "10"}, {"_value": "12.50"}, {"_value": "11.00"}, {"_value": "9.25"}, {}]}, {"td": [{"_value": "11"}, {"_value": "13.75"}, {"_value": "12.25"}, {"_value": "10.50"}, {}]}, {"td": [{"_value": "12"}, {"_value": "15.00"}, {"_value": "12.00"}, {"_value": "11.75"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "16.25"}, {"_value": "13.25"}, {"_value": "13.00"}, {"_value": "11.25"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "_value": "best price"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}]}, {"td": [{"_value": "6"}, {"_value": "6.00"}]}, {"td": [{"_value": "7"}, {"_value": "7.25"}]}, {"td": [{"_value": "8"}, {"_value": "8.50"}]}, {"td": [{"_value": "9"}, {"_value": "8.00"}]}, {"td": [{"_value": "10"}, {"_value": "9.25"}]}, {"td": [{"_value": "11"}, {"_value": "10.50"}]}, {"td": [{"_value": "12"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "11.25"}]}]}]}]}
Ask students,
“’Number of bagels’ and ’price’ do not form a
function
, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
“If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
“If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”
After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles Jada’s price, red squares Priya’s price, and yellow triangles Han’s price. In the second graph, each X represents the best price for each number of bagel.
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Because there are multiple possible prices when the number of bagels is 6 or greater, price is not a function of the number of bagels bought.​​​​​​
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Because there is only one best price for a particular number of bagel, best price is a function of the number of bagels bought.
Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river.”).


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