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ALGEBRA I

Hart Interactive – Algebra 1 M4 Lesson 3

Lesson 3: Investigating the Parts of a Parabola

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 symmetry? 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 y-axis?

G. Where does the graph cross the x-axis?

H. What is lowest point on the graph?

Lesson 3: Investigating the Parts of a Parabola Unit 10: Introduction to Quadratics and Their Transformations

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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015

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ALGEBRA I


To learn all the math vocabulary associated with parabolas, we’ll watch a YouTube video and look at some pictures of parabolas.

2. Watch the YouTube video, Quadratic Functions and Parabolas in the Real World at https://www.youtube.com/watch?v=He42k1xRpbQ&list=PLGM5-h0pP9-9-8iBUJszxZ7jkXftY99s-.

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.

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•

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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 symmetry does.

Axis of Symmetry


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ALGEBRA I


7. The pictures below refer to the vertex of a parabola. What do they mean by vertex?

8. The graph at the right, illustrates the location of the x-intercepts and the y-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 y = (x + 1)(x – 3) or y = x2 – 2x – 3 or y = (x – 1)2 – 4.

9. Go back to the Opening Exercise and fill in the correct mathematical vocabulary in the last column of the

chart. Then mark and label the vertex, axis of symmetry, x-intercepts and y-intercept on the graph.


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ALGEBRA I


In Exercise 8B, you looked at three ways to write the quadratic equation of the parabola. All three forms of the quadratic equation give very 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 identify the vertex of the parabola. Then use the Equation Bank on the next page to find the equation that matches each graph.

10.

11.

Vertex: ____________

Equation: _____

Vertex: ____________ Equation: _____

12.

13.

Vertex: ____________

Equation: _____

Vertex: ____________ Equation: _____


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ALGEBRA I


14.

15.

Vertex: ____________ Equation: _____

Vertex: ____________ Equation: _____

16.

17.

Vertex: ____________

Equation: _____

Vertex: ____________ Equation: _____

EQUATION BANK

A. 2( 1) 2y x= − − B. 2( 2) 1y x= − + + C. 2( 1) 4y x= − + D. 2( 4) 1y x= − +

E. 2( 0) 4y x= + + F. 2( 2) 3y x= + − G. 2( 4) 0y x= − + + H. 2( 3) 1y x= − − −

18. What did you notice about the equations for the parabolas in Exercises 14 – 17?


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ALGEBRA I


Use the equation frame to write an equation of a parabola that has the given vertex.

19. Vertex: (3, -1) Equation: y = (x ___ _____)2 + ____

20. Vertex: (-4, 0) Equation: y = (x ___ _____)2 + ____

21. Vertex: (0, -6) Equation: y = (x ___ _____)2 + ____

22. Vertex: (2, 4) Equation: y = (x ___ _____)2 + ____

23. Vertex: (-5, -7) Equation: y = (x ___ _____)2 + ____

Lesson Summary

AXIS OF SYMMETRY: The axis of symmetry 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 symmetry intersect is called the vertex.

END BEHAVIOR OF A GRAPH: Given a quadratic function in the form 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐 (or 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 −ℎ)2 + 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, 𝑦𝑦 = 𝑎𝑎(𝑥𝑥 − ℎ)2 + 𝑘𝑘, (ℎ,𝑘𝑘) are the coordinates of the vertex.


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ALGEBRA I


Homework Problem Set

Below are some examples of curves found in architecture around the world. Some of these might be represented by graphs of quadratic functions.

1. What are the key features these curves have in common with a graph of a quadratic function? Mark each key feature on the picture.

St. Louis Arch Bellos Falls Arch Bridge

Arch of Constantine Roman Aqueduct

2. How would you describe the overall shape of a graph of a quadratic function?

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•

• 3. What is similar or different about the overall shape of the above curves?


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ALGEBRA I


4. Below you see only one side of the graph of a quadratic function.

A. Complete the graph by plotting three additional points of the quadratic function. Explain how you found these points, and then fill in the table on the right.

B. What are the coordinates of the 𝑥𝑥-intercepts?

C. What are the coordinates of the 𝑦𝑦-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 your completed graph from Problem 4A to verify that the average rate of change for the interval −3 ≤ 𝑥𝑥 ≤ −2, or [−3,−2], is 5. Show your steps.

6. Based on your work in Problem 5, what interval would have an average rate of change of −5? Explain your thinking.

𝒙𝒙 𝒇𝒇(𝒙𝒙)


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