Section 8.2 Quadratic Functions and Their Graphs
Definition Quadratic Function
A quadratic function is a second-degree polynomial function of the form
,
where a, b, and c are real numbers and
. Every quadratic function has a “u-shaped” graph
called a parabola.
Objective 1: Identify the characteristics of a quadratic function from its graph
A parabola either opens up or opens down depending on the leading coefficient, . If
, as in
Figure 1a, the parabola will “open up.” If
, as in Figure 1b, the parabola will “open down.”
If
If
, the graph will be narrower than the graph of
, the graph will be wider than the graph of
.
8.2.1 Without graphing, determine if the graph of the quadratic function opens up or down. Also
determine if the graph will be wider or narrower than the graph of
Five basic characteristics of a parabola:
1.
2.
3.
4.
5.
Vertex
Axis of symmetry
y-intercept
x-intercept(s) or real zeros
Domain and range
8.2.5 Use the given graph of a quadratic function to find
a. Vertex
b. Axis of symmetry
c. y-intercept
d. x-intercept(s)
e. Domain and range
Objective 2: Graph quadratic functions by using translations
In mathematics, a translation is when every point on a graph is shifted the same distance in the same
direction. We now examine translations of parabolas involving vertical or horizontal shifts.
Vertical Shifts of Functions
If c is a positive real number:
The graph of
The graph of
is obtained by shifting the graph of
is obtained by shifting the graph of
vertically upward k units.
vertically downward k units
Horizontal Shifts of Functions
If c is a positive real number:
The graph of
The graph of
is obtained by shifting the graph of
is obtained by shifting the graph of
horizontally left h units.
horizontally right h units
8.2.11 Sketch the graph of the quadratic function _______________ by using translations. Compare
the graph to the graph of
.
Objective 3: Graph quadratic functions of the form f ( x)  a  x  h   k
2
Standard Form of a Quadratic Function
A quadratic function is in standard form if it is written as f ( x)  a  x  h   k .
2
The graph is a parabola with vertex (h, k ) .
xh
xh
(h, k ) 
a0
Domain:  ,  
Range:
 (h, k )
k, 
a0
Domain:  ,  
Range:
 , k 
The axis of symmetry of the parabola is the vertical line x = h.
Given the quadratic function in standard form f ( x)  a  x  h   k , answer the following:
2
1.
2.
3.
4.
5.
6.
7.
What are the coordinates of the vertex?
For what values of a does the graph open up?
What is the equation of the axis of symmetry?
How do you find any x-intercepts?
How do you find any y-intercept?
State the domain.
State the range.
Open down?
8.2.15 Given the quadratic function in standard form, find 1-7 above.
Objective 4: Find the vertex of a quadratic function by completing the square
Writing
in Standard Form by Completing the Square
Step 1. Group the variable terms together.
Step 2. If
, factor a out of the variable terms.
Step 3. Take half the coefficient of the x-term inside the parentheses, square it, and add it inside the
parentheses. Multiply this value by a, then subtract from c.
Step 4. The expression inside the parentheses is now a perfect square. Rewrite it as a binomial
squared and simplify the constant term outside of the parentheses.
8.2.19 Write the quadratic function in standard form and find the vertex.
8.2.21 Write the quadratic function in standard form and find the vertex.
Objective 5: Graph quadratic functions of the form
by completing the square.
Rewrite the function in standard form by completing the square, then answer/do the following
1.
2.
3.
4.
5.
6.
7.
8.
What are the coordinates of the vertex?
Does the graph open up?
Open down?
What is the equation of the axis of symmetry?
x-intercepts?
y-intercept?
Graph
State the domain.
State the range.
8.2.24 Rewrite the quadratic function _________________ in standard form then answer 1-8 above.
8.2.27 Rewrite the quadratic function _________________ in standard form then answer 1-8 above.
Objective 6: Find the vertex of a quadratic function by using the vertex formula
Formula for the Vertex of a Parabola
Given a quadratic function of the form
,
given by
.
The axis of symmetry is the vertical line
.
, the vertex of the parabola is
8.2.31 Use the vertex formula to find the vertex of the quadratic function ________________
Objective 7: Graph quadratic functions of the form
formula
Given the quadratic function
, answer the following
1. What are the coordinates of the vertex?
2. Does the graph open up?
Open down?
3. What is the equation of the axis of symmetry?
4. x-intercepts?
5. y-intercept?
6. Graph
7. State the domain.
8. State the range.
8.2.34 Use the quadratic function __________________ to answer 1-8 above.
by using the vertex