Math 1314
3.1 Quadratic Functions
Section 3.1 Notes
Vertical Parabolas
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Math 1314
Quadratic Functions:
Section 3.1 Notes
Any equation written in the form y = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0 is said to define a
quadratic function.
General form: y = f(x) = ax2 + bx + c for a ≠ 0
Standard form: y = f(x) = a(x – h)2 + k for a ≠ 0
Graph of Quadratic Function in Standard Form y = f(x) = a(x – h)2 + k for a ≠ 0
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Math 1314
Section 3.1 Notes

a=

Vertex:

Axis of symmetry:

y-intercept:

x-intercept(s):

Additional points:
Note: Parabolas can be graphed by transformations of functions.
Note:
-
If |a| > 1, the parabola is narrower (vertical stretching from y = x2)
-
If |a| < 1, the parabola is wider (vertical shrinking from y = x2)
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Math 1314
Section 3.1 Notes

a=

Vertex:

Axis of symmetry:

y-intercept:

x-intercept(s):

Additional points:
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Math 1314
Section 3.1 Notes
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Math 1314
Section 3.1 Notes

a=

Vertex:

Axis of symmetry:

y-intercept:

x-intercept(s):

Additional points:
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Math 1314
Example 4 : Graph the quadratic function f(x) = x2 + 6x + 8 in general form.

a=

Vertex:

Axis of symmetry:

y-intercept:

x-intercept(s):

Additional points:
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Section 3.1 Notes
Math 1314
Section 3.1 Notes
Steps for writing a quadratic function f(x) = ax2 + bx + c in standard form.
1. Group the variable terms f(x) = (ax2 + bx) + c
2. Factor out the coefficient a of the terms in x2 and x: f ( x )  a  x 2  x   c
b
a 

.
2
b
3. Add and subtract   inside the parentheses to complete the square inside the parentheses, we obtain
 2a 

b
b2
b2 
f ( x )  a  x 2  x  2  2   c
a
4a
4a 

4. Now we are ready to complete the square.
2
b
b2
Add and subtract    2

b
b2
b2 
f ( x)  a x 2  x  2  2   c
a
4a
4a 

 2a 

b
b2  b2
f ( x )  a  x 2  x  2  
c
a
4a  4 a

b2
Multiply 4a 2 by coefficient a then move it outside
2
Then the first three terms form a perfect square
b  b2

f ( x)  a x   
c
2a  4a

x2 
Then we obtain the function in the standard form
2
b 
b2

f ( x)  a x    c 
2a 
4a

4a
2
where h   b ; k  c  b
2a
Example: Rewrite the function in standard form f ( x)  3x 2  12 x  17 .
f ( x)  3x 2  12 x  17
 3x 2  4 x   17
 3x 2  4 x  4  4  17
 3x 2  4 x  4  12  17
 3x  2  5
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This is standard form with a = 3, h = 2 and k =5.
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4a
b
b2 
b 
x  2 x  
a
4a
2a 

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Math 1314
Section 3.1 Notes
Examples: Write each quadratic function from general to the standard form, then find the vertex, extreme value
of the function.
1. y = f(x) = x2 – x – 20
2. y = f(x) = – 2x2 – x + 6
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Math 1314
Section 3.1 Notes
3. y = f(x) = ½ x2 + x – 3
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