Int. Alg. Notes
Section 8.5
Page 1 of 8
Section 8.5: Graphing Quadratic Equations Using Properties
Big Idea: There are formulas that convert between the general form and standard form of a quadratic function.
Big Skill: You should be able to use those formulas to convert between forms so that you can quickly sketch the
graph of a quadratic function
Quadratic function in general form: f  x   ax2  bx  c
Quadratic function standard form: f  x   a  x  h   k
Instead of completing the square every time we are given a quadratic function to graph, we can complete the
square on the general form of the quadratic function, and thus get formulas for h and k.
2
Completing the square on the general form of a quadratic function:
 Make sure the coefficient of the square term is 1.
f  x   ax 2  bx  c

b 

f  x   a  x2  x   c
a 

Identify the coefficient of the linear term; multiply it by ½ and square the result.
2

b2
1 b
    2
4a
2 a
Add that number to both sides of the equation. Don’t forget the factor of a that distributes onto the number
you are adding… Also notice that this step is the same thing as adding and subtracting the same number
on the right hand side.
b 

f  x   a  x2  x   c
a 

f  x  a 
 2 b
b2
b2 

a
x

x


c
4a 2
a
4a 2 


b
b2 
b2
f  x   a  x2  x  2   c 
a
4a 
4a

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes

Section 8.5
Page 2 of 8
Write the resulting perfect square trinomial as the square of the binomial.
 2 b
b2 
b2
f  x  a  x  x  2   c 
a
4a 
4a

2
b  4ac b 2

f  x  a  x   

2a 
4a 4a

b  4ac  b 2

f  x  a  x   
2a 
4a

Compare the completed square to the standard form to identify h and k.
2
b  4ac  b 2

f  x  a  x   
2a 
4a

2

2
  b   4ac  b 2
f  x  a  x     
4a
  2a  
f  x  a  x  h

2
h
b
2a
k
4ac  b 2
k
4a
The Vertex of a Parabola
Any quadratic function in general form f  x   ax2  bx  c (a  0) will have its vertex at the point whose
coordinates are:
 b 4ac  b2 
 ,
 .
4a 
 2a
Two alternative ways to state the vertex coordinates are using the discriminant:
D
 b
D  b 2  4ac    ,  
 2a 4a 
And by plugging the x-coordinate of the vertex into the function (i.e., since y = f(x) ):
 b
 b 
  2a , f   2a  



Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 3 of 8
Practice:
1. Compute the coordinates of the vertex of the parabola specified by the quadratic
function f  x   3x2  7 x  1.5 .
The x-Intercepts of the Graph of Parabola
The x-intercepts of a graph are the x values where y = 0:
y0
f  x  0
ax 2  bx  c  0
Thus, the x-intercepts of the graph of a parabola are given by the quadratic formula. We can anticipate the
number of x-intercepts based on the discriminant:
If the discriminant D  b 2  4ac  0 , then the graph of f  x   ax2  bx  c has two different x-intercepts at
b  D
.
2a
f  x   2 x 2  5x  1
x
D  b 2  4ac
D  52  4  2  1
D  33
x
b  D
2a
5  33
x

4
5  33
 0.186
4
5  33

 2.686
4

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 4 of 8
If the discriminant D  b 2  4ac  0 , then the graph of f  x   ax2  bx  c has one x-intercept, and the vertex of
the graph will touch the x-axis at x 
f  x   1.7 x2  6.8x  6.8
b
.
2a
D  b 2  4ac
D  6.82  4  1.7  6.8 
D0
b  D
2a
6.8  0
x
2  1.7 
x
x2
If the discriminant D  b 2  4ac  0 , then the graph of f  x   ax2  bx  c has no x-intercepts (the graph does
not cross or touch the x-axis).
f  x   0.6x2  2.1x  3.6
D  b 2  4ac
D  2.12  4  0.6  3.6 
D  4.23
x
b  D
2a
2.1  4.23i
2.1  4.23
1.2
x

1.2
2.1  4.23i

1.2

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 5 of 8
Practice:
2. Compute the x-intercepts of the parabola specified by the quadratic function f  x   3x2  7 x  1.5 .
3. Compute the x-intercepts of the parabola specified by the quadratic function f  x   2 x2  12 x  18 .
4. Compute the x-intercepts of the parabola specified by the quadratic function f  x   x 2  2 x  2 .
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 6 of 8
To Graph a Quadratic Function Using Its Properties:
b
4ac  b 2
 Use the formulas h  
and k 
to quickly convert the general form of the quadratic equation,
2a
4a
2
f  x   ax2  bx  c , to the standard form f  x   a  x  h   k .
 Graph the standard form using translations.
Practice:
5. Sketch a graph of y  x 2  2 x  3 using its properties.
6. Sketch a graph of y  x 2  10 x  25 using its properties.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 7 of 8
7. Sketch a graph of y  2 x 2  6 x  7 using its properties.
Application of Graphing Quadratic Functions: The vertex of a quadratic function is either the max or min
value of the function.
Practice:
8. A company’s daily revenue R as a function of the price of its product p is given by:
1
R  p    p 2  300 p . Find the price that maximizes the daily revenue and the maximum revenue.
2
9. A farmer has 2000 feet of fencing to enclose a rectangular field. Find the maximum area that can be
fenced off and the dimensions of that maximum size field.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.5
Page 8 of 8
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.