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
$f(x)=-3{x}^{2}-12x-16$

**Analysis:**

The function is a quadratic function so the graph is a parabola which opens up or down.

Because the leading coefficient is negative the parabola opens down.

The fact that the discriminant
${b}^{2}-4ac={\left(-12\right)}^{2}-\left(4\right)\left(-3\right)\left(-16\right)=144-192<0$
is negative, implies the parabola does not have an x-intercept.

such that it does not intersect the parabola.

The vertex is an important point and should be marked

Because exact values cannot be inferred from a sketch it is important that we label the exact values for the vertex.
Use the fact that the vertex is
$\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)=\left(\frac{12}{2(-3)},f\left(\frac{12}{2(-3)}\right)\right)=\left(-2,f\left(-2\right)\right)=\left(-2,-4\right)$

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**

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 -4. The range is
$[-4,-\infty )$

The graph clearly shows the answer to

and because
$f(x)=-3{x}^{2}-12x-16$
the answer to Where is
$f(x)=0$? is also the solution set for the equation in one variable
$-3{x}^{2}-12x-16=0$.

The graph clearly shows the answer to

and because
$f(x)=-3{x}^{2}-12x-16$
the answer to Where is
$f(x)<0$? is also the solution set for the inequality in one variable
$-3{x}^{2}-12x-16<0$.

The graph clearly shows the answer to

and because
$f(x)=-3{x}^{2}-12x-16$
the answer to Where is
$f(x)>0$? is also the solution set for the inequality in one variable
$-3{x}^{2}-12x-16>0$.

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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