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
$f(x)=-{x}^{2}-10x-25=(-1)({x}^{2}+10x+25)=(-1){\left(x+5\right)}^{2}$

**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
${b}^{2}-4ac={\left(-10\right)}^{2}-(4)(-10)(-25)=100-100=0$
shows that it is zero.

Therefore there is one x-intercept.

This same information is apparent from the fact that $0=f(x)={\left(x-3\right)}^{2}$ 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
$[0,-\infty )$

The graph clearly shows the answer to

and because
$f(x)=-{x}^{2}-10x-25$
the answer to Where is
$f(x)=0$?
is also the solution set for the equation in one variable
$-{x}^{2}-10x-25=0$.

The graph clearly shows the answer to

and because
$f(x)=-{x}^{2}-10x-25$
the answer to Where is
$f(x)<0$?
is also the solution set for the inequality in one variable
$-{x}^{2}-10x-25<0$.

The graph clearly shows the answer to

and because
$f(x)=-{x}^{2}-10x-25$
the answer to Where is
$f(x)>0$?
is also the solution set for the inequality in one variable
$-{x}^{2}-10x-25>0$.

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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