Download Simplifying Square Roots

Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Investigation
Part I: Create a scatter plot of the solutions to the quadratic function
f ( x)  x 2  5 x  6 on your graphing calculator.
(i) Create a table of solutions of f ( x) by following the steps below.



Clear all numbers from all lists by pressing the “2nd “ key followed
by the “+” key and choosing option 4. Press enter.
Enter all integers between –8 and 8 in L1 by pressing STAT and
choosing EDIT.
Calculate the corresponding Y values and put them in L2 by entering
the function L2  L1  5L1  6 . Note that to do this you must move
your cursor to the column header “ L2 ” and press enter before you
enter the formula. Press enter.
Observe the range of values for Y and X and set your window to
include these values. Use Xscl  Yscl  1.
2

(ii) Plot the ordered pairs ( x, y ) on a graph using STAT PLOT 1. Use the
“  ” to mark the points.
(iv) Convince yourself that all of the solutions of f ( x) must lie somewhere
on this curve by adding several more values. You can make these values
as messy as you like! ( i.e {1.5213,  , 5, (1  2),...} . Note that you
must recalculate L2 each time after you add numbers to L1 by re-entering
the function L2  L1  5L1  6 .
2
(v) Graph the equation f ( x)  x 2  5 x  6 using the graphing utility on your
calculator. What is the difference between your scatter plot and the
graph you obtained by using the graphing utility? What is the domain
and range of f ( x) ?
Part II:
(i) Use algebra to find the coordinates of the x and y intercepts in the graph of
f ( x)  x 2  5 x  6 .
(ii) Describe the symmetry in the graph of f ( x) . Using the symmetry in the
graph determine the coordinates of the vertex (the lowest point on the graph.)
(iii) Use the intersect function to solve the equation x 2  5 x  6  25 (round your
answers to three decimal places.) What happens when you try to use algebra
to solve this equation?
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Notes…
The ordered pairs representing the solutions of f ( x)  ax 2  bx  c all lie on a curve
called a parabola.
Some characteristics of a parabola and hence the graph of f ( x)  ax 2  bx  c are :
 The graph has a minimum or a maximum point called the vertex.
 The graph is symmetric about a horizontal line passing through the vertex.
The equation of this line of symmetry is given by x  (the x coordinate of the
vertex).
 The x coordinate of the vertex is equal to the average value of the x
coordinates of any 2 points on the parabola having the same y coordinate.
 The shape of the parabola has many reflective properties that are used
frequently in real life. Some examples are automobile headlights, satellite
dishes, etc…
Procedure for graphing f ( x)  ax 2  bx  c when f ( x) can be factored (i.e.
expressed in intercept form.)
Example: Sketch the graph of f ( x)  6 x 2  15 x  9 .
Step 1: Factor f ( x) completely.
f ( x)  6 x 2  15 x  9
f ( x)  3(2 x 2  5 x  3)
f ( x)  3(2 x  1)( x  3)
Step 2: Use the zero product property to determine the x intercepts (the zeros of
f(x). Also, determine the y intercept.
f ( x)  0
3(2 x  1)( x  3)  0
2 x  1  0 or x  3  0
1 
 The coordinates of the x intercepts are  , 0 
2 
and (-3,0)
The y intercept = f (0)  6(0)2  15(0)  9  9
 The coordinates of the y intercept are (0,9)
Step 3: Determine the coordinates of the vertex. Also, state the equation of the
axis of symmetry.
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
The x coordinate of the vertex = the average of the x intercepts =
1

 3   / 2  5 / 4
2

The y coordinate of the vertex is given by
 5
 5 
 5 
f     6    15    9  18.375
 4
 4 
 4 
The coordinates of the vertex are  -1.25,18.375 
2
and the equation of the axis of symmetry is x  -1.25.
Step 4: Sketch the graph of f ( x) . Be sure to use a straightedge, sharpened pencil,
and show the values of the intercepts and vertex and state the equation of the
axis of symmetry. Also, state the domain and range of f ( x) .
The next part of our study of quadratics will focus upon finding the exact solutions to
quadratic equations of the form ax 2  bx  c  0 particularly in the cases where
ax 2  bx  c cannot be factored. As we will see soon these solutions are often
irrational and can be expressed using square roots. So before proceeding we need to
take a few moments to review square roots and simplifying square roots.
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Definition: The square root of a number x is a number such that when squared gives
the value n. In other words,
x is the square root of n  x 2  n
Considerations…

2 is the square root of 4 because 22  4 . (Note that even though –2 also
satisfies the definition… (2)2  4 it is common practice to take only the
positive answer…and is referred to as the principal square root.)

Be careful though…the solutions to the equation x 2  9 are x   9 or
x  3 .

The square roots of non – perfect squares are irrational numbers. That is, they
are numbers that cannot be expressed as fractions (ie. That are non-repeating,
non-terminating decimals). For example, 8  2.82842712...

The square roots of negative numbers are said to be non-real. You will see in
pre-calculus that these answers do in fact exist and have many applications in
real-life situations!
The Real Number System
The Real Number system is made up of two basic types of numbers: The
rational numbers and the irrational numbers.
REAL NUMBERS: 
IRRATIONALS: I
RATIONALS: ¤
(cannot be written as fractions)
(can be written as fractions)
 Square roots of non perfect squares
 Numbers like  and e
Integers : ¢
Whole Numbers: W
Natural Numbers: ¥
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Properties of square roots (assuming a, b  0 )




a  b  ab
 a  a    a   
2

a2  a
a
a

b
b
p a  q a  ( p  q ) a (square roots can be added provided that the
radicands are the same number)
Simplifying square roots
An expression containing one or more square roots is said to be simplified if all of the
following conditions are met:
 No radicand contains a factor that is a perfect square…if they do we need to
simplify using a  b  ab
 No radicand contains a fraction…if they do we need to simplify using
a
a
.

b
b
 All fractions have denominators that do not contain square roots…if they do
we need to simplify using a
a  a . This process is called rationalizing
  
the denominator.
Some examples…Simplify each square root.
(ii) 2 3  6
(i) 5 27  4 45




(iv) 4 3
5
(vii) 2  3 2  3
(v)


3
5
(vii)
(iii)
3 5
2

4
9
(vi) 2 3  3 5
3
2 7

37 5

Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Example : Solve 3( w  5)2  24 .
3( w  5) 2  24
( w  5) 2  8
w5  8
w52 2
3
4
Example: 1  ( x  3)2  7
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
Assignment
1.
Throughout our discussions in class we have referred to f ( x)  ax 2  bx  c as the
general quadratic function. Is it correct to call this a function or should we call
this relation? Support your answer using the definition of a function.
2.
Sketch a graph of the solutions to each quadratic function below using the 4 step
process outlined in class.
(a) W (a)  a 2  5a  24
(b) S ( p)  3 p 2  2 p  5
f ( x)  9  4 x 2
(d) h(r )  8r 2  8r  2
(c)
3.
Give an example of a quadratic function that can be expressed in intercept form
as f ( x)  a( x  h)2 . What can be said about the number of zeros of such a
function? What does the graph of the function look like? State the coordinates of
the x and y intercepts and vertex in terms of a and h.
4.
Find the principal square root of each number. Round your answer to 3 decimal
places.
(a) 11
5.
(b) 85
(c) -3
2
3
(d)
Simplify each expression. Express your answers in exact simplified form.
(a) 3 28
(b) 7 108
(e) 4 3  18
(f)
(i) (2  4)(3  5)
(m)
5 3 
(j)
3-2 3+5 5-4 3

(c)
3
40
4
(g)
 2 6 5 3 
2
32 5
(n)

2
2 8
+2 2
3
(k) 3
(o)
(d)
2
3
2 3
3 2 2
2
128
3
(h)
 2
(l)
2
1 3
3
Graphing Quadratic Functions in Intercept Form
and
Simplifying Square Roots
7.
8.
Solve each quadratic equation. Express your answers in exact simplified form.
Verify your answers using a calculator.
(a) c 2  8  0
(b) 3v 2  7  8
(c) (r  6)2  20
(d) (k  5)2  1  2
(e) 4(2s  3)2  5  11
(f) 13  (4 x  5)2  2
Consider the quadratic function f ( x)  ( x  1)2  5 .
(a) Using algebra find the x and y intercepts of the graph. Express your answers
in exact simplified form.
(b) Using algebra find the coordinates of the vertex.
(c) Sketch a graph of your equation. Label the exact coordinates of the
intercepts and vertex on your graph. Also show the line of symmetry for
your graph along with its equation.
9.
List all the sets of numbers for which each number below belongs to. (i.e.
Natural, Whole, Integer, Rational, Irrational, or Real)
(a) 1.5
(b)
7
(c) 2 16
(d) . 1
10. Find the area of a rectangle that has a width of 3 cm and diagonals whose lengths
are 54cm . (Hint: Use must first use the Pythagorean Theorem)