### DrDelMathGraphing a Quadratic FunctionwithNegative 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.

Problem:Analyze the quadratic function $f\left(x\right)=-{x}^{2}+2x+3$

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 ${b}^{2}-4ac={2}^{2}-\left(4\right)\left(-1\right)\left(3\right)=4+12>0$ shows that it is positive.
Therefore there are two x-intercepts.
such that it intersects the parabola in two points.
Because the x-intercepts are important for any graph we should .
Because exact values cannot be inferred from a sketch it is important that we calculate the exact values for the x-intercepts.
The x-intercepts of the graph of any function $f$ are the real solutions of the equation resulting from $f\left(x\right)=0$. These real solutions are also called real zeros of the function $f$. In this example we must solve the equation $-{x}^{2}+2x+3=0$ which is equivalent to (multiply both sides by -1) ${x}^{2}-2x-3=0$. Because ${x}^{2}-2x-3=\left(x+1\right)\left(x-3\right)$ we can use factoring and The Zero Factor Property to determine that the solutions to the equation are $-1$ and $3$. Both are real numbers so they correspond to x-intercepts. On any graph the x-intercepts should always be labeled with their coordinates.

The vertex of a parabola is also an important point on the graph of a quadratic function so we should
The vertex should be labeled with its coordinates. The vertex is $\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)=\left(\frac{-2}{2\left(-1\right)},f\left(\frac{-2}{2\left(-1\right)}\right)\right)=\left(1,f\left(1\right)\right)=\left(1,4\right)$
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 Interpreting the Graph
The graph shows that the domain is all real numbers R
The graph shows that the range is all real numbers greater less or equal to 4. The range is $\left[4,-\infty \right)$

The graph clearly shows the answer to
and because $f\left(x\right)=-{x}^{2}+2x+3$ the answer to Where is $f\left(x\right)=0$ ? is also the solution set for the equation $-{x}^{2}+2x+3=0$.

The graph clearly shows the answer to
and because $f\left(x\right)=-{x}^{2}+2x+3$ the answer to Where is $f\left(x\right)<0$ ? is also the solution set for the inequality $-{x}^{2}+2x+3<0$.

The graph clearly shows the answer to
and because $f\left(x\right)=-{x}^{2}+2x+3$ the answer to Where is $f\left(x\right)>0$ ? is also the solution set for the inequality $-{x}^{2}+2x+3>0$.

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 $\left(a,f\left(a\right)\right)$ where $a$ is a domain element and $f\left(a\right)$ 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 $f\left(a\right)$. That fact together with the observations related to points in the plane permits the following.
A point $\left(a,f\left(a\right)\right)$ on the graph of $f$ is above the x-axis if and only if $f\left(a\right)>0$.
A point $\left(a,f\left(a\right)\right)$ on the graph of $f$ is on the x-axis if and only if $f\left(a\right)=0$.
A point $\left(a,f\left(a\right)\right)$ on the graph of $f$ is below the x-axis if and only if $f\left(a\right)<0$.
.