Lesson 3: Investigating the Parts of a Parabola
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- Eleanore Wilson
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1 Opening Exercise 1. Use the graph at the right to fill in the Answer column of the chart below. (You ll fill in the last column in Exercise 9.) Question Answer Bring in the Math! A. What is the shape of the graph? B. Does this graph have smmetr? Explain. C. What happens when x > -3? D. What happens when x < -3? E. What are some of the special points on this graph? F. Where does the graph cross the -axis? G. Where does the graph cross the x-axis? H. What is lowest point on the graph? Unit 10: Introduction to Quadratics and Their Transformations S.15
2 To learn all the math vocabular associated with parabolas, we ll watch a YouTube video and look at some pictures of parabolas.. Watch the YouTube video, Quadratic Functions and Parabolas in the Real World at Then answer the questions or fill in the blanks below. 3. The graph of a quadratic functions forms a -shape, called a. 4. Give three examples of parabolas in the real world. 5. What is a problem that can be solved with a quadratic function? 6. Use the picture at the right to tell what the axis of smmetr does. Axis of Smmetr Unit 10: Introduction to Quadratics and Their Transformations S.16
3 7. The pictures below refer to the vertex of a parabola. What do the mean b vertex? 8. The graph at the right, illustrates the location of the x-intercepts and the -intercept. A. Describe what intercepts are on a graph. B. How can we find them if we have the equation of a parabola? In this case the equation of this parabola can be written as = (x + 1)(x 3) or = x x 3 or = (x 1) Go back to the Opening Exercise and fill in the correct mathematical vocabular in the last column of the chart. Then mark and label the vertex, axis of smmetr, x-intercepts and -intercept on the graph. Unit 10: Introduction to Quadratics and Their Transformations S.17
4 In Exercise 8B, ou looked at three was to write the quadratic equation of the parabola. All three forms of the quadratic equation give ver specific information about the graph of the parabola. In the next exploration, we ll focus on one of those forms. Exploration Focus on the Vertex For each graph below identif the vertex of the parabola. Then use the Equation Bank on the next page to find the equation that matches each graph Unit 10: Introduction to Quadratics and Their Transformations S.18
5 EQUATION BANK A. = ( x 1) B. = ( x+ ) + 1 C. = ( x 1) + 4 D. = ( x 4) + 1 E. = ( x+ 0) + 4 F. = ( x+ ) 3 G. = ( x+ 4) + 0 H. = ( x 3) What did ou notice about the equations for the parabolas in Exercises 14 17? Unit 10: Introduction to Quadratics and Their Transformations S.19
6 Use the equation frame to write an equation of a parabola that has the given vertex. 19. Vertex: (3, -1) Equation: = (x ) + 0. Vertex: (-4, 0) Equation: = (x ) + 1. Vertex: (0, -6) Equation: = (x ) +. Vertex: (, 4) Equation: = (x ) + 3. Vertex: (-5, -7) Equation: = (x ) + Lesson Summar AXIS OF SYMMETRY: The axis of smmetr is a vertical line through a parabola so that each side is a mirror image of the other side. VERTEX: The point where the graph of a quadratic function and its axis of smmetr intersect is called the vertex. END BEHAVIOR OF A GRAPH: Given a quadratic function in the form ff(xx) = aaxx + bbxx + cc (or ff(xx) = aa(xx h) + kk), the quadratic function is said to open up if aa > 0 and open down if aa < 0. VERTEX FORM: When graphing a quadratic equation in vertex form, = aa(xx h) + kk, (h, kk) are the coordinates of the vertex. Unit 10: Introduction to Quadratics and Their Transformations S.0
7 Homework Problem Set Below are some examples of curves found in architecture around the world. Some of these might be represented b graphs of quadratic functions. 1. What are the ke features these curves have in common with a graph of a quadratic function? Mark each ke feature on the picture. St. Louis Arch Bellos Falls Arch Bridge Arch of Constantine Roman Aqueduct. How would ou describe the overall shape of a graph of a quadratic function? 3. What is similar or different about the overall shape of the above curves? Unit 10: Introduction to Quadratics and Their Transformations S.1
8 4. Below ou see onl one side of the graph of a quadratic function. A. Complete the graph b plotting three additional points of the quadratic function. Explain how ou found these points, and then fill in the table on the right. xx ff(xx) B. What are the coordinates of the xx-intercepts? C. What are the coordinates of the -intercept? D. What are the coordinates of the vertex? Is it a minimum or a maximum? E. If we knew the equation for this curve, what would the sign of the leading coefficient be? 5. Use our completed graph from Problem 4A to verif that the average rate of change for the interval 3 xx, or [ 3, ], is 5. Show our steps. 6. Based on our work in Problem 5, what interval would have an average rate of change of 5? Explain our thinking. Unit 10: Introduction to Quadratics and Their Transformations S.
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