Averaging the Intercepts
Averaging the Intercepts
A. Graphing and Standard Form
1. Use the graphing calculator to verify that the two equations [pic] and [pic]are equivalent.
2. Show algebraically that these two equations are equivalent by starting with the graphing form and showing step by step how to get the standard form.
3. Notice that the value for a is 3 in both forms of the equation, but that the numbers for b and c are different from the numbers for h and k. Why do you think the value for a would be the same in both forms of the equation?
B. Ms. Speedi, who teaches Algebra knew she would not be at class on Friday because she would be attending a mathematics conference. Before leaving school Thursday, she wrote on the board a list of 10 parabola equations in graphing form, with instructions. The instructions said:
TEAM QUIZ
Use what you learned yesterday about the graphing form of quadratic equations to sketch the graphs of these parabolas without using your graphing calculators.
On Friday morning the substitute arrived late, and the custodian let the first period Algebra I class into the room. Several eager students saw the assignment on the board and thought it was for them. They immediately went to the board and started doing the one thing they had learned so far, multiplying out the squared expression and simplifying. They had just finished replacing the last equations with its simplified form when the substitute arrived with a completely different assignment for them.
When the students in the fourth period Algebra 2 class arrived, they read the instructions and saw the list of equations, all in standard form. The first one was: [pic].
Making the xy-tables and plotting point-by-point would take forever. They had to find another way. Enrico said, “We need that other form of the equation that tells us the vertex. How do we get that?”
1. Jessica suggested, “We know how to get the coordinates of the x-intercepts.” What are the coordinates of the x-intercepts for this equation?
2. Kevin said, “And parabolas are symmetric, so the vertex is mid-way between them.” How could you find the x-value of the vertex? Find it.
3. Kenya volunteered, “And if we know the x-value, we can find the y-value that goes with it.” Find the y-value.
4. “We’ve got the vertex. Now we can make a sketch!” the all said together. Make a sketch of this function, label the intercepts and the vertex, and draw in a line of symmetry.
5. “How can we be sure we’re right?” asked Jessica. “Couldn’t we write the graphing form of the equation then work it out to see if we get the original one in standard form?” Enrico suggested. Use the vertex to write the equation in graphing form. Show your work to verify that it is the same as the original by squaring and simplifying.
C. The following four functions were also part of Ms. Speedi’s Quiz. For each function do parts (1) through (5) as in the previous problem in order to write and verify the graphing form:
1. [pic]
2. [pic]
3. [pic]
4. [pic].
D. On Friday afternoon the custodian erased all the boards, so when Ms. Speedi returned on Monday, she assumed her Algebra 2 students had just started with the graphing forms. She never realized that her class had figured out their own method for changing the equation from standard form to graphing form.
Based on your work with the two preceding problems, describe the students’ method for finding the vertex without drawing the graph and for figuring out the graphing form of the equation of a parabola from the standard form. With your team, make up another example, and show how to use the method you described.
E. Find the vertex of each of the following parabolas by averaging the x-intercepts. Then write each equation in graphing form.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
F. Did you need to average the x-intercepts to find the vertex in part (4) of the preceding problem?
1. What are the coordinates of the vertex in part (4)?
2. How do these coordinates relate to the equation?
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