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.
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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 and r2 , 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  15
. )( x  35
.)
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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.
i)
ii)
6. The roots for a quadratic function are given. Write the equation of each function.
i)
ii)
iii)
iv)
r1  2, r2  3
r1  2.5, r2  5
r1  6, r2  1
r1  1, r2  8
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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
. )( x  35
.)
iv) m( x )   ( x  15
3
Compare these graphs to the ones you made in #1. What happens to the roots of
f ( x)  a( x  r1 )( x  r2 ) when a  1 ?
Part 2 – General Form
Definition: The general form of a quadratic function is given as: f ( x)  ax 2  bx  c ,
where a, b, & c are real numbers.
8. Graph each of the following on a graphing calculator.
i) t ( x)  x 2  3x  10
ii) r ( x)  x 2  14 x  13
iii) u( x)  x 2  11x  28
iv) v( x)  x 2  2 x  5.25
9. Identify the roots for each of the above equations.
i)
ii)
iii)
iv)
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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  35
. ) 2  0.25
iv)
i ( x)  ( x  15
. )2  4
v)
j( x) 
1
( x  2) 2  6
2
vi) k ( x)  1( x  7.5) 2  3.25
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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.
i)
ii)
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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.
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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???????
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Project AMP
Dr. Antonio R. Quesada, Director Project AMP
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)  x 2  6x  5
 ( x 2  6 x  ??)  5  ??
F6 I F6 I
 6 x  GJ)  5  GJ
H2 K H2 K
2
 (x
2
2
 ( x 2  6 x  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)  x 2  4 x  10
b) h( x)  2 x 2  4 x  13
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