DrDelMath
Graphing a Quadratic Function
with
Negative Leading Coefficient and Discriminant = 0

This is a demonstration of how I anticipate using HTML, CSS, JavaScript, SVG, mathML, and mathJax to provide meaningful online instruction in basic algebra topics as found in Intermediate Algebra and College Algebra.

Example 1:Analyze the quadratic function f whose rule is given by f(x)= x210x 25 =(1)( x 2 +10x+25)=(1) (x + 5 ) 2

Analysis:
The function is a quadratic function so the graph is a parabola which opens up or down.
Because the leading coefficient is positive the parabola opens up.

A quick examination of the discriminant b24ac= ( - 10 ) 2 (4)( 10)(25)=100100=0 shows that it is zero.
Therefore there is one x-intercept.
This same information is apparent from the fact that 0=f( x)= (x3) 2 has only one solution. Which means the parabola has only one x-intercept.
such that it intersects the parabola in one point.
We should also observe that the x-intercept must be the vertex. Therefore the vertex and x-intercepts have the same coordinates (-5,0). The x-intercept/vertex should be marked
Because exact values cannot be inferred from a sketch it is important that we label the vertex with exact values for its coordinates.
The graph of the function f is now complete.
All the important information about the function is summarized and displayed in geometic form.


Interpreting the Graph

The graph shows that the domain is all real numbers R
The graph shows that the range is all real numbers less than or equal to 0. The range is [0,-)

The graph clearly shows the answer to
and because f(x)= x210x25 the answer to Where is f(x )=0? is also the solution set for the equation in one variable x2 10x25=0.

The graph clearly shows the answer to
and because f(x)= x210x25 the answer to Where is f(x )<0? is also the solution set for the inequality in one variable x2 10x25<0.

The graph clearly shows the answer to
and because f(x)= x210x25 the answer to Where is f(x )>0? is also the solution set for the inequality in one variable x2 10x25>0.

 

 

A Few Observations About the Interpretation
Definition:The graph of a function f consists of all the points and only the points of the form (a,f(a)) where a is a domain element and f(a) is the unique range element associated with a.

In the coordinate plane a point is above the x-axis if and only if its second coordinate is positive (greater than 0).
In the coordinate plane a point is on the x-axis if and only if its second coordinate is equal to 0.
In the coordinate plane a point is below the x-axis if and only if its second coordinate is negative (less than 0).
When we compare second coordinates to 0 The Law of Trichotomy yields three cases (greater than 0, equal to 0, and less than 0) and those three cases match up nicely with geometric properties (above, on, or below the x-axis) of the Cartesian Coordinate System.

The definition of graph of a function informs us that the second coordinate of every point on the graph is of the form f(a). That fact together with the observations related to points in the plane permits the following.
A point (a,f(a)) on the graph of f is above the x-axis if and only if f(a)>0.
A point (a,f(a)) on the graph of f is on the x-axis if and only if f(a)=0.
A point (a,f(a)) on the graph of f is below the x-axis if and only if f(a)<0.
.