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.

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

**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
${b}^{2}-4ac={2}^{2}-(4)(-1)(3)=4+12>0$
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
$f(x)=0$.
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
$\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)=\left(\frac{-2}{2(-1)},f\left(\frac{-2}{2(-1)}\right)\right)=\left(1,f\left(1\right)\right)=\left(1,4\right)$

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
$[4,-\infty )$

The graph clearly shows the answer to

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

The graph clearly shows the answer to

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

The graph clearly shows the answer to

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

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 $f\left(a\right)<0$.

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