Finding the Vertex of a Quadratic Function



Lesson Plan

1. Lesson Title: Finding the Vertex of a Quadratic Function

2. Lesson Summary: This is an inquiry-based lesson designed to teach students how to find the vertex of a quadratic function. The lesson will teach students how to find the vertex using different methods including: table of values, graphing and using minimum/maximum tools on a graphing calculator, algebraic method for standard form and vertex form. At the end of the lesson, students will use the knowledge that they have gained to solve real world problems involving quadratic functions.

The lesson is divided into the following four sections: Graphic Methods, Standard Form, Vertex Form and Applications. Teachers could use some or all of the sections individually.

All quadratics have integral values for the coordinates of the vertex and can be viewed in the standard window of most graphing calculators. This was done so that students can focus on the concepts rather than the details of finding the vertex. Changing coefficients so that vertex coordinates are not integral and that the vertex cannot be found in the standard window could easily increase the difficulty of the lesson as well as using more complex applications. Also, converting between forms of a quadratic function are not addressed in the lesson.

Each activity would be one day for a standard period classroom (40-50 minutes). Block scheduled classes could complete the assignment in one day.

3. Key Words: Vertex, Quadratic, Maximum, Minimum.

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4. Background knowledge: The students should be able to use a graphing calculator to perform the following tasks: input equation, graph, examine table of values and find the minimum or maximum value in range. Students need to be able to evaluate a quadratic function for a given value. Parts of the lesson could be completed without the use of a graphing calculator.

The material covered would be appropriate for Algebra 1 and Algebra 2 students of varying abilities.

5. Ohio Standards Addressed:

Indicators:

Grade 10, Patterns, Functions and Algebra

10. Solve real-world problems that can be modeled using linear, quadratic, exponential, or square root functions.

Grade 11, Patterns, Functions and Algebra

3. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions.

4. Identify the maximum and minimum points of polynomial, rational and trigonometric functions, graphically and with technology.

Benchmarks:

Grade 10: D: Use algebraic representations, such as tables, graphs, expressions, functions, and inequalities to model and solve problem situations.

Grade 11: A: Analyze functions by investigating rates of change, intercepts,

zeroes, asymptotes, and local and global behavior.

6. Learning Objectives: Students will understand the concept that the vertex is either the minimum or maximum value of a quadratic function. Students will be able to find the vertex of a quadratic using a variety of methods including: table of values, graphing and using minimum/maximum tools on a graphing calculator, algebraic method for standard form and vertex form. Student will be able to apply this knowledge to solve real world problems involving quadratic functions.

7. Materials: Graphing calculator. This was written for use with a TI-83 or TI-84. Some directions would need to be modified for other graphing calculators.

8. Suggested procedures:

a. The instructor can choose an “attention getter” of his/her choice. Some examples could be “How do you find the maximum height of bottle rocket?” “How do you find the maximum height of Lebron James when he is dunking a basketball?” Also, any other question involving projectile motion that would interest your specific groups of students could be used.

b. Students could work individually or they could be grouped in any manner that the instructor sees fit to use. It is suggested that students work cooperatively and then review the lesson as a larger group.

9. Assessment(s): A variety of assessments could be used to evaluate students’ knowledge of the material. It is suggested that both formative and summative assessments are implemented. Each section has a problem that allows students to test their conjectures and/or findings. These could be used as formative assessments. In terms of summative assessments, this lesson in and of itself would probably be best included in a larger assessment.

SOLUTIONS

Finding the Vertex of a Quadratic Function

Activity 1. Graphic Methods

(Note: all graphs shown were graphed on TI-Smartview and pasted into the document. It is understood that students will most likely be doing these sketches by hand after they have graphed the quadratic on their calculators.)

1. Graph the following functions on your graphing calculator. Sketch the graph below. Every quadratic function will have either a maximum or minimum value. Determine whether each function has a maximum or minimum value.

f(x) = x2 – 3 g(x) = 2x2 – 4x

Minimum Minimum

h(x) = -3x2 – 12x + 1 k(x) = - x2 + 8x – 16

Maximum Maximum

2. Make a conjecture about how you could determine whether a quadratic function has either maximum or minimum value WITHOUT graphing?

If the lead coefficient is positive the function will have a minimum value.

If the lead coefficient is negative the function will have a maximum value.

Students could also refer to the coefficient of x2. If students refer to the graph opening up or down, remind them that the question is asking how you can determine if there is a max or min WITHOUT graphing.

3. Determine if the following functions have either maximum or minimum values. Then check your answer by graphing. Include a sketch of your graph. (You do not need to find the actual value of the max/min)

a(x) = -2x2 – 3x - 2 b(x) = 5x2 – 4x – 9

Maximum Minimum

c(x) = -x2 + 2x + 1 j(x) = x2 + 8x - 10

Maximum Minimum

4. Was your conjecture correct? If not, what would be a more accurate conjecture?

Answers will vary.

Define Vertex: The vertex of a quadratic function can be defined in many ways. For our purposes right now, we will define the vertex as the maximum or minimum point of the graph of the quadratic.

5. Find the vertex of each of the following functions by graphing the function on your graphing calculator. (Remember: Go to 2nd-calc-minimum/maximum. Now you must select a value to the left and right of the vertex. Then hit Enter until the calculator gives you the value at the vertex.) Include a sketch of the graph.

Students should know how to find the maximum/minimum value using the graphing calculator. If this method has not yet been taught the teacher will want to give greater detail, including examples, for how to do this.

r(x) = -2x2 – 4x - 2 s(x) = 4x2 + 8x - 6

(-1,0) (-1,-10)

t(x) = x2 – 10x + 22 w(x) = -3x2 – 12x – 7

(5,-3) (-2,5)

6. . Now for each of the functions in number 5, complete the table of values below. Include the x-value at the vertex (place this in the box marked V) and the next three x-values greater than and less than the x-value at the vertex.

(Note: all tables shown were created on TI-Smartview and pasted into the document. It is the teacher’s discretion as to whether students can go to the table function on their graphing calculators and copy values or if they must calculate the y-values. Also, the teacher may want to fill in the x-values before handing out the worksheet or provide the x-values for the students as they work.)

r(x) = -2x2 – 4x - 2 s(x) = 4x2 + 8x - 6

t(x) = x2 – 10x + 22 w(x) = -3x2 – 12x – 7

7. What do you notice about the table of values? (hint: think symmetry)

The values are symmetric about the vertex. As each x-value increases or decreases by one, the y value changes by the same amount. You could show this graphically as well. This explanation could take on many forms depending on the level of the class.

An extension would be to have students find missing x or y values from the table using the symmetric properties of a quadratic function.

8. Make a conjecture about how you could determine the vertex of a quadratic function based on a table of values?

The vertex is the coordinate with either the highest or lowest y value. The vertex is the coordinate about which the rest of the graph is symmetric. Again conjectures and their detail will vary depending on the ability of the class.

9. From the table of values below, determine the vertex of each quadratic function. Decide if it is a maximum or a minimum. Also connect this to your conjecture in question #4 and determine the sign of the coefficient of the x2 term.

|x |y |

|-2 |8 |

|-1 |3 |

|0 |0 |

|1 |-1 |

|2 |0 |

|3 |3 |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

Minimum at (1,-1) Minimum at (1,0)

|x |y |

|1 |-4 |

|2 |1 |

|3 |4 |

|4 |5 |

|5 |4 |

|6 |1 |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

Maximum at (4,5) Maximum at (–8,-5)

10. Was your conjecture correct? If not, what would be a more accurate conjecture?

Answers will vary.

11. Review and Compare: For each function below find the vertex using both methods that we have discussed so far:

A. Graph and use max/min function on your calculator. Sketch the graph.

B. Graph and examine table of values. Show at least 5 table values.

Again, the answers were created using TI-Smartview.

This question could be used as a type of assessment (formative or summative) to determine the students’ level of comprehension before continuing on to Activity 2.

q(x) = x2 – 6x + 4 p(x) = -2x2 - 8x – 6

vertex (3,-5) Minimum vertex (-2,2) Maximum

m(x) = -x2 + 10x - 15 n(x) = 3x2 – 12x +3

vertex (5,10) Maximum vertex (2,-9) Minimum

Activity 2. Algebraic Methods – Standard Form

Definition of Standard form of a quadratic function: The standard form of a quadratic function is when the quadratic is written as f(x) = ax2 + bx + c

12. Place each of the following functions in standard form and identify the values for a, b and c. (these are the same functions that we evaluated in Activity 1, Number 5 just in a different format)

Teachers may want to skip this question and just refer to the functions in #5 if students will not be required to rewrite quadratics in standard form or if the instructor believes the students are already proficient on this topic. The section may also be expanded if the teacher believes that more complexity and/or practice is needed.

r(x) = -2 - 2x2 – 4x s(x) = 4x2 + 6x – 6 + 2x

2x2 – 4x – 2 4x2 + 8x – 6

t(x) = x2 + 22 – 10x w(x) = -2x2 – 22x – 7 – x2 + 10x

x2 – 10x + 22 -3x2 – 12x – 7

13. Find the value of negative b divided by 2 times a (–b/2a) for each of the functions in number 12.

Work should be shown for each function.

r(x) = -1

s(x) = -1

t(x) = 5

w(x) = -2

14. What do you notice about the x-values you found in the vertices of the functions in #5 and the values of –b/2a in number 13?

The x-values for the vertices in #5 are equal to –b/2a that we calculated in #13. Try to get students to answer beyond just “the x’s are all the same”.

15. Take the value of –b/2a that you found for each function in #13 and plug that back into the original function for x and evaluate the function.

Work should be shown for each function.

r(x) = 0

s(x) = -10

t(x) = -3

w(x) = 5

16. What do you notice about the y-values you found in the vertices of the functions in #5 and the values you found in #15?

The y-values for the vertices in #5 are equal to what is obtained by evaluating the function for –b/2a in #15. Try to get students to go beyond “the y’s are all the same”.

17. Make a conjecture about the relationship between the vertex of a quadratic function and the values of a, b and c in the standard form of a quadratic equation.

The vertex is (–b/2a, f(-b/2a)) also: x=-b/2a, y=f(-b/2a) also: x=-b/2a, y=what you get when you plug in x. Again the notation and detail depends on the level of the class.

Students answers may include the fact that c has no impact on the x-value of vertex. An extension may be a discussion on why c (constant) has no impact on the x-value. Make sure that students are aware that c does impact the vertex because it is include in the calculation of the y-value at the vertex.

Other extensions for higher ability/level student may include tying -b/2a for the vertex to the quadratic formula and the symmetry of the equation and zeroes.

18. Write a step by step algebraic process (algorithm) that someone could use to find the vertex of a quadratic function that is in standard form.

Answers will vary – example below

1. Put equation in standard form

2. Identify a, b, c

3. Find –b/2a to find x at the vertex

4. Plug this into the function and you get y at the vertex

19. Use the algorithm that you have created to find the vertex of each of the following quadratic functions. Verify your answers by using one of the graphing methods you used in Activity 1. Show your work below.

Teachers may or may not want students to include a verifying graph. Student work should match the algorithm that they created in #18.

f(x) = 4x2 + 8x – 3

vertex (-1,-7)

g(x) = ½ x2 + 8x +25

vertex (-8,-7)

h(x) = -4x2 + 24x – 31

vertex (3,5)

i(x) = -x2 - 5

vertex (0,5)

20. Was your algorithm from #18 correct? If not, what would be a more accurate algorithm? Answers will vary.

Activity 3. Algebraic Methods – Vertex Form

Definition of Vertex form of a quadratic function: The vertex form of a quadratic function is when the quadratic is written as f(x) = a(x-h)2 + k

21. Each of the following functions are in vertex form. Identify the values for a, h and k. (these are the same functions that we evaluated in Activity 1, Number 5 just written in a different form) (Hint: the formula is –h not h, so remember to take the opposite of what you see in the function.)

Teachers may want to spend more time explaining and/or reviewing the fact that students need to take the opposite of h that is seen in the function.

If the transformations of functions have already been discussed, this question could be tied to horizontal and vertical translations. This could also be a point where an extension to the lesson takes place involving horizontal and vertical translations.

r(x) = -2(x+1)2 s(x) = 4(x+1)2 - 10

a=-2 a=4

h=-1 h=-1

k=0 k=-10

t(x) = (x-5)2 – 3 w(x) = -3(x + 2)2 + 5

a=1 a=-3

h=5 h=-2

k=-3 k=5

22. What do you notice about the x-values you found in the vertices of the functions in #5 and the values of h in #21?

The x-values are the same as h. Make sure that students understand that the formula is (x-h) and that h is the opposite of the value in the parenthesis of the function.

23. What do you notice about the y-values you found in the vertices of the functions in #5 and the values of k in number #21?

The y-values are the same as k.

24. Make a conjecture about the relationship between the vertex of a quadratic function and the values of a, h and k in the vertex form of a quadratic equation.

The vertex of the quadratic function written in standard form is given by (h, k). Also, x at the vertex equals the opposite of what is in parenthesis and y at the vertex equals what is added/subtracted outside of the parenthesis. Again the complexity required for the answer depends on the ability level of the students.

25. Write a step by step algebraic process that someone could use to find the vertex of a quadratic function that is written in vertex form.

Answers will vary. An example is below:

1. find h and k, h is the opposite of what you see in parenthesis

2. The vertex is at (h, k)of the quadratic function written in standard form is given by (h, k)

26. Use the algorithm that you have created to find the vertex of each of the following quadratic functions. Verify your answers by using one of the graphing methods you used in Activity 1.

Students should show work according to what they have written in #25, for this particular problem, they may have little or no work.

f(x) = -2(x-6)2 + 8 g(x) = (x+2)2 - 3

(6,8) (-2,-3)

q(x) = 2(x – 3)2 p(x) = -3(x + 5)2 -1

(3,0) (-5,-1)

27. Was your conjecture correct? If not, what would be a more accurate conjecture?

Answers will vary

Activity 4. Applications

28. An object thrown into the air is modeled by the equation: h(t) = -4.9t2 + vot + ho where vo is the initial velocity, ho is the initial height above the ground and h(t) is the height after t seconds.

A person throws a ball with an initial upward velocity of 12 m/sec. The ball is released when it is 1.8 meters from the ground.

A. Find the equation that would give the height of the ball t seconds later.

h(t) = -4.9t2 + 12t + 1.8

B. After how many seconds will the ball reach its peak (i.e. maximum height)?

What will the maximum height be?

Use at least two methods. Show your work below.

time=1.22 seconds, maximum height =9.15 meters

The best methods are probably are graphing and using the max utility or finding –b/2a and then plugging this value back in for maximum height.

The teacher may want to take this opportunity to discuss why these are the two best methods and how appropriate methods should be chosen.

29. Kevin wants to fence in his backyard using a rectangular region so that his dogs can have a safe play area. He has purchased 140 feet of fencing. To help the fencing cover a greater square area, Kevin will use the back of his house as one side of the enclosed yard.

A. Find the equation that would give the area of the rectangular region that Kevin fenced in. Include a drawing to show how you determined the equation.

There are two answers. One with the width=x and one with the length=x

Answer with width = x

140-2x

x x

The dimensions are x and 140-2x.

The equation would be f(x)=x(140-2x) or –2x2 + 140x

Answer with length = x

The dimensions are x and ½(140-x).

The equation would be f(x)=x(½( 140-x) or 70x– ½x2

x

½ (140-x) ½ (140-x)

The teacher may want to discuss both answers and explain why they are both valid, why the vertices are different, etc.

A. Find the maximum area that can be enclosed.

Find the dimensions that produce the maximum area.

Use at least two methods. Show your work below.

The dimensions are 35 feet for the sides and 70 feet for the one long side.

The maximum area is 2450 square feet.

The teacher may want to discuss how the answer was obtained based on each method shown above. For this problem the algebraic method of –b/2a was appropriate. Graphing with max/min technology would be difficult due to the large numbers unless students were familiar with changing the window values and even then it could be problematic. Using the table may have been easier and faster than graphing. Again a discussion of which method would have been best may be worthwhile at this point.

30. Quad Hardware Company produces c-clamps. The production cost will go down the more units the company produces. However, you also know that production costs will eventually go up if you make too many c-clamps, due to storage requirements and overtime pay. The equation to produce x thousands of c-clamps a day can be approximated by the following formula:

C = 0.08x2 – 8.5x + 2600 cost is in thousands of dollars.

A. Find the daily production level of c-clamps that will minimize production costs for Quad Hardware.

B. Find the minimum production cost.

Production level is 53125 c-clamps and the production cost is $2,374,218.75

The actual vertex for the function is (53.125, 2374.21875). Students need to recognize that both x and y are in thousands in the equation.

The best method for this problem would be to solve algebraically using –b/2a. The table is problematic since the x-value at the vertex is a decimal value. Graphing represents the same difficulties that were found in #29.

This problem also presents the issue of what is a possible answer. If students do not realize that x is in thousands of units, they may answer 53 for x and use that value to find the minimum value of the function.

An extension of the problem would be to find the average cost per c-clamp.

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|x |y |

|-2 |27 |

|-1 |12 |

|0 |3 |

|1 |0 |

|2 |3 |

|3 |12 |

|x |y |

|-11 |-14 |

|-10 |-9 |

|-9 |-6 |

|-8 |-5 |

|-7 |-6 |

|-6 |-9 |

house

house

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