Teaching Notes 8.13: Describing the Graph of the Quadratic
Function
The quadratic equation provides information about the vertex and axis of symmetry of the graph
of the function. Some students have trouble organizing and relating all of the information needed
to describe the graph.- Review the squaring function,f(x)=x^2 , with your students. Explain that it is a parabola
that opens upward. The opposite of this function,g(x)=−x^2 , is a reflection of the parabola
in thex-axis. It opens downward. Note that the graphs of all squaring functions are
parabolas. - Explain thatf(x)=ax^2 +bx+cis the general form of the squaring function. Ifa>0, the
graph is a parabola that opens upward. Ifa<0, the graph opens downward. - Explain that all parabolas have a vertex, which is the lowest point of the graph if it opens
upward or the highest point if it opens downward. By looking at the equation, students
should be able to determine thex-coordinate of the vertex, which is−
b
2 a. Students can then
substitute this value into the equation and solve for they-coordinate of the vertex.
4. Explain that every parabola has an axis of symmetry, a line where a parabola can be folded so
that each side of the graph will coincide. The equation of the axis of symmetry isx=−
b
2 a
.
- Review the information and example on the worksheet with your students. You may wish to
sketch the graph by plotting points to verify the vertex and axis of symmetry.
EXTRA HELP:
The line of symmetry will always intersect the vertex of the parabola.ANSWER KEY:
(1)Minimum point(
1
2
,
7
4
)
;x=1
2
(2)Maximum point(
− 3
2
,
13
4
)
;x=− 3
2
(3)Minimum point (0,−1);x= 0 (4)Maximum point(
3
2
,−
9
2
)
;x=3
2
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(Challenge)Ifa=0 was substituted into the quadratic equation,y=ax^2 +bx+c,theresulting
equation would bey=bx+c, which is a linear equation.
------------------------------------------------------------------------------------------298 THE ALGEBRA TEACHER’S GUIDE