You will develop the competencies: communicating in English, mathematical and basic scientific‑technical and IT competence, the learning skills.
You will learn what the vertex form of the quadratic function is.
Learning effect
You plot a graph of the function .
You determine the properties of the function based on its graph.
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Remind how the function formulaformulaformula changes when the graph is shifted along the OX axis and along the OY axis.
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If you shift the graph of functionfunctionfunction by:
units to the right along the OX axis then you get the graph of the functiongraph of the functiongraph of the function
units to the left along the OX axis then you get the graph of the function
units up along the OY axis then you get the graph the function
units down along the OY axis then you get the graph of the functiongraph of the functiongraph of the function
Task 1
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Find a formulaformulaformula of the function which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units to the left along the axis OX.
Find a formulaformulaformula of the functionfunctionfunction which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units up along the axis OY.
Task 2
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Find a formulaformulaformula of the function which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units to the left along the axis OX.
Find a formulaformulaformula of the functionfunctionfunction which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units up along the axis OY.
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Using the results of your work determine the formulaformulaformula of the functionfunctionfunction which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units along the OX axis and units along the OY axis.
Formulate the conclusion.
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Shifting the graph of the functionshifting the graph of the functionShifting the graph of the function , where , by units along the OX and units along the OY axis results in a graph of the functiongraph of the functiongraph of the function . The formulaformulaformula is called the vertex formvertex formvertex form of a quadratic equation.
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Apply the acquired knowledge in exercises.
Task 3
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Determine the formulaformulaformula of the functionfunctionfunction which graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units to the left along the OX axis and then units down along the OY axis.
Task 4
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The functionfunctionfunction graph was obtained as a result of shifting the graph of the functionshifting the graph of the functionshifting the graph of the function by units along the OX axis and units along the OY axis. Find the numbers , , .
Task 5
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Open the applet and observe how the and numbers in the quadratic functionquadratic functionquadratic function formula vary, depending on the change of the coordinates of the parabola's vertex, which is the graph of this functionfunctionfunction. Draw the conclusion.
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The parabola ist the graph of the functiongraph of the functiongraph of the function , where . The parabola is congruent to the graph of the functionfunctionfunction . The parabola’s vertex is the point .
Task 6
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Sketch the graph of a given functionfunctionfunction and discuss its properties.
Task 7
An extra task:
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Determine the formulaformulaformula of the quadratic functionquadratic functionquadratic function f in the vertex formvertex formvertex form assuming that for the argument the functionfunctionfunction has the maximum valuevaluevalue and the point lies on the graph.
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Perform consolidating exercises.
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Students perform consolidating exercises. Then they summarize together the activities, formulating conclusions to be remembered:
Shifting the graph of the functionshifting the graph of the functionShifting the graph of the function , where , by units along the OX and units along the OY axis results in a graph of the functiongraph of the functiongraph of the function .
The formulaformulaformula , where is known as the vertex formvertex formvertex form of a quadratic functionquadratic functionquadratic function.
The ordered pair is the vertex of the parabolavertex of the parabolavertex of the parabola , where .
Exercises
Exercise 1
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Exercise 2
A graph of a quadratic function f was obtained as a result of shifting the function . The function f is increasing function in the interval and decreasing function in the interval . The maximal value of this function is the number . Determine the function formula f in the vertex form.
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Exercise 3
Describe in English the properties of the function .