Team 2 Lesson: the Three Forms of Quadratic Functions

Project AMP Dr. Antonio R. Quesada, Director Project AMP

Three Forms of a Quadratic Function

This lesson is designed for the learner to work between the three forms of a quadratic function: general, vertex, and factored. Through an inquiry format, the student will use a graphing calculator to aid them in writing equations and switching between the different forms.

Key Words: quadratic equations, vertex form, factoring quadratics, completing the square

Students should be able to multiply two binomials, know the definition of a zero and root of an equation, and be able to graph quadratic equations by hand and on the graphing calculator before they have this lesson.

Ohio State Standard: Patterns, Functions, and Algebra Standard Benchmarks:  8-10C- Translate information from one representation to another representation of a relation or function.  8-10E- Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeroes.  8-10G- Solve quadratic equations with real roots by graphing, formula and factoring.

Objectives:  Through using the graphing calculator, the student will be able to use the factored form of the quadratic equation to find the roots of the function. The student will be able to find the factored form of the quadratic equation from given roots and graph.  The student will be able to find the vertex of a parabola from the general form of the quadratic equation with a graphing calculator and then write the equation in the vertex form of the quadratic equation.  The student will be able to write an equation of all three forms (general, vertex, factored) for one parabola and give the importance of each form.

Materials: graphing calculator, graph paper, writing instrument, ruler, brain.

Procedures: Students may work alone or in pairs.

Assessment: Assessment and extension problems.

- 1 - Project AMP Dr. Antonio R. Quesada, Director Project AMP

Three Forms of a Quadratic Function

Lesson

This lesson is designed to help the student learn how to convert and compare the three forms of quadratic functions.

Part 1 – Factored Form

Definition: Recall that when a function crosses the x-axis, those x-values are said to be roots. For a quadratic function that has real roots, r1 andr2 , the factored form is given as:

f (x)  a(x  r1)(x  r2 ) ; where a is a real number.

Example: f (x)  (x 1)(x  5)

1. Without the use of a calculator, make a table of values to sketch the graph for each of the following.

i) f (x)  (x  5)(x  2) ii) g(x)  (x 1)(x 13) iii) h(x)  (x  7)(x  4) iv) i(x)  (x 1.5)(x  3.5)

2. Graph each function from #1 on a graphing calculator to confirm your results.

3. How do the roots relate to the equations of the functions?

4. The roots of a quadratic function are often referred to as the zeroes of the function. Why do you think this is?

5. Write the function of each graph given below.

6. The roots for a quadratic function are given. Write the equation of each function.

i) r1  2, r2  3

ii) r1  2.5, r2  5

iii) r1  6, r2  1

iv) r1  1, r2  8

7. Now use a graphing calculator and consider the graphs of:

i) j(x)  2(x  5)(x  2) ii) k(x)  2(x 1)(x 13) 2 iii) l(x)   (x  7)(x  4) 3 2 iv) m(x)   (x 1.5)(x  3.5) 3

Compare these graphs to the ones you made in #1. What happens to the roots of

f( x )= a ( x - r1 )( x - r 2 ) whena  1?

Part 2 – General Form

Definition: The general form of a quadratic function is given as: f (x)  ax2  bx  c , where a, b, & c are real numbers.

8. Graph each of the following on a graphing calculator. i) t(x)  x2  3x 10 ii) r(x)  x2 14x 13 iii) u(x)  x2 11x  28 iv) v(x)  x2  2x  5.25

9. Identify the roots for each of the above equations.

10. How do the roots in #9 compare with the roots of functions in #1?

11. Based on your observations, how can you convert the equations from #1 into those given in #8? Check your hypothesis with CAS on a TI-89 calculator.

Part 3 – Vertex Form

Definition: Another form of a quadratic function, known as vertex form, is given as: f (x)  a(x  h)2  k , where h & k are real numbers.

Let’s take a closer look at this form. Remember the vertex of a parabola is either the minimum or the maximum point of the function.

12. Graph each of the following functions and identify the vertex of each using the calculator.

i) f (x)  (x  2)2  3

ii) g(x)  (x  4)2  5

iii) h(x)  (x  3.5)2  0.25

iv) i(x)  (x 1.5)2  4

1 v) j(x)  (x  2)2  6 2

vi) k(x)  1(x  7.5)2  3.25

13. How does the vertex of each function relate to the equation in vertex form of the function?

14. How does the value of a affect the vertex of each parabola?

15. Write the equation for each function given a and the vertex.

i) a  1, vertex: (5, 4)

ii) a  2 , vertex: (-2, -7)

16. Write the equation of each function shown in the graphs below.

17. List all the definitions and the properties that you learned in this lesson.

Extension 1

A model rocket blasts off from a position 2.5 meters above the ground. Its starting velocity is 49 meters per second. Assume that it travels straight up and that the only force acting on it is the downward pull of gravity. In the metric system, the acceleration due to gravity is 9.8 m/sec2. 1 The quadratic function h(t)  (9.8)t 2  49t  2.5 describes the rocket’s projectile motion. 2 a) Graph the function h(t) on your graphing device.

b) How high does the rocket fly before falling back to Earth? When does it reach this point?

c) How much time passes while the rocket is in flight?

d) When is the rocket 50 meters above the ground? Use a calculator table to approximate your answers to the nearest hundredth of a second.

e) Describe how to answer (d) graphically.

f) Write the equation of this quadratic function in vertex form.

Extension 2

a) Write the equation of the following parabola in vertex form (assume a = 1).

b) Now, convert the vertex form of the equation into general form. Show your work.

c) Write the equation in factored form.

i. Problems????????

ii. Why???????

Extension 3

Converting from vertex form into general form may not be much of a challenge for you. Let’s consider changing general form back to vertex form? This might be a bit more difficult for you.

In order to convert general form into vertex form, we need a technique called completing the square. An example is given below.

f (x)  x2  6x  5  (x2  6x  ??)  5 ?? 6 2 6 2  (x2  6x  GFJI )  5 GFJI H2K H2K  (x2  6x  9)  5 9  (x  3)2  4

Now, you try…

Convert the general form of the function into vertex form. Check your equation using your calculator.

a) g(x)  x2  4x 10

b) h(x)  2x2  4x 13

SOLUTIONS:

1. i) ii)

iii) iv)

2. See solutions to number 1

3. The roots are the opposite of the values given in the function.

4. They are called zeroes because the roots are where the graph crosses the x-axis where y = 0.

5. i) f (x)  (x  4)(x  3) ii) g(x)  (x 1)(x  4.5)

6. i) f (x)  (x  2)(x  3) ii) g(x)  (x  2.5)(x  5) iii) h(x)  (x  6)(x 1) iv) i(x)  (x 1)(x  8)

7. In factored form, the a value does not affect the roots of the function.

8. i) ii)

iv) iv)

9. i) r1  5, r2  2; vertex  (1.5, 12.25)

ii) r1  13, r2  1; vertex  (7,  36)

iii) r1  7, r2  4; vertex  (5.5,  2.25)

iv) r1  1.5, r2  3.5; vertex  (1,  6.25)

10. The roots in #9 are the same as the roots in #1.

11. Student answers may vary. You can FOIL the functions in #1 to get the functions in #8.

12. i) ii)

iii) iv)

v) vi)

13. The x-value of the vertex is the opposite of the number in the parentheses. The y- value of the vertex is the number that comes after the parentheses.

14. The value of a does not affect the vertex of the parabola.

15. i) y  (x  5)2  4 ; ii) y  2(x  2)  7

16. i) y  (x  3)2  2 ; ii) y  (x  4)2  3

17. Students should define/comment on some or all of the following: a. Factored Form of a Quadratic Function

i. For a quadratic function that has real roots, r1 and r2 , the factored

form is given as: f (x)  a(x  r1)(x  r2 ) b. General Form of a Quadratic Function i. The general form of a quadratic function is given as: f (x)  ax2  bx  c c. Vertex Form of a Quadratic Function i. Vertex form is given as: f (x)  a(x  h)2  k d. Roots of a function i. When a function crosses the x-axis, those x-values are said to be roots. ii. Roots are also known as zeroes of a function. e. How a affects the roots and the vertex of a function i. The value of a neither affects the roots nor the vertex when in factored form and vertex form.

Extension 1 a) window: [0, 15, 1, 0, 150, 10]; x-axis: time (sec), y-axis: distance (meters)

b) 125 feet after 5 seconds

c) 10 seconds

d) The rocket is 50 meters above the ground at about 1.09 seconds and at about 8.91 seconds.

e) Graph y  h(t) and y  50 on the same set of axes. Find the points of intersection using the intersect function on the calculator.

1 y  (9.8)(t  5)2 125 f) 2  4.9(t  5)2 125

Extension 2 y  (x  0)2  4 a)  x2  4

b) y  x2  4

c) Cannot be written in factored form because the function has no roots. The graph never crosses the x-axis. This means that x2  4 cannot be factored.

Extension 3

a) g(x)  x2  4x 10 4 2 4 2  (x2  4x  GFJI ) 10  GFJI H2K H2K  (x2  4x  4) 14  (x  2)2 14

b) h(x)  2x2  4x 13 2 2 2 2  2(x2  2x  GFJI 13 GFJI H2K H2K  2(x2  2x 1) 14  2(x 1)2 14

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