## The Discriminant and the Quadratic Graph

The discriminant of the quadratic equation $ax^2+bx+c=0$ is $\Delta = b^2-4ac$.We used the discriminant to determine the number of real roots of the quadratic equation. If they exist, these roots correspond to $x$-intercepts of the quadratic $y = ax^2+bx+c$. The discriminant tells us about the relationship between a quadratic function and the $x$-axis.

Case 1: $y=x^2-2x+2$

\( \begin{align} \displaystyle a &= 1, b=-2, c=2 \\ \Delta &= b^2-4ac \\ &= (-2)^2 - 4 \times 1 \times 2 \\ &= -4 \lt 0 \\ \end{align} \)

The graph does not cut the $x$-axis and is always above the $x$-axis, which means $y=x^2-2x+2 \gt 0$.

This case is Positive Definite because $x^2-2x+2$ is always positive for all real $x$.

Case 2: $y=x^2-2x+1$

\( \begin{align} \displaystyle a &= 1, b=-2, c=1 \\ \Delta &= b^2-4ac \\ &= (-2)^2 - 4 \times 1 \times 1 \\ &= 0 \\ \end{align} \)

The graph touches the $x$-axis the graph is mostly above the $x$-axis, which means $y=x^2-2x+1 \ge 0$.

This case is Not Positive Definite because $x^2-2x+1$ is not always positive for all real $x$.

Case 3: $y=x^2-2x$

\( \begin{align} \displaystyle a &= 1, b=-2, c=0 \\ \Delta &= b^2-4ac \\ &= (-2)^2 - 4 \times 1 \times 0 \\ &= 4 \gt 0 \\ \end{align} \)

The graph cuts the $x$-axis twice and and the graph is above or below the $x$-axis depending on $x$ values.

This case is Not Positive Definite because $x^2-2x$ is not always positive for all real $x$.

Case 4: $y=-x^2+2x-2$

\( \begin{align} \displaystyle a &= -1, b=2, c=-2 \\ \Delta &= b^2-4ac \\ &= 2^2 - 4 \times (-1) \times (-2) \\ &= -4 \lt 0 \\ \end{align} \)

The graph does not cut the $x$-axis and is always below the $x$-axis, which means $y=-x^2+2x-2 \lt 0$.

This case is Negative Definite because $-x^2+2x-2$ is always negative for all real $x$.

Case 5: $y=-x^2+2x-1$

\( \begin{align} \displaystyle a &= -1, b=2, c=-1 \\ \Delta &= b^2-4ac \\ &= 2^2 - 4 \times (-1) \times (-1) \\ &= 0 \\ \end{align} \)

The graph touches the $x$-axis the graph is mostly below the $x$-axis, which means $y=-x^2+2x-1 \le 0$.

This case is Not Positive Definite because $x^2-2x+1$ is not always negative for all real $x$.

Case 6: $y=-x^2+2x$

\( \begin{align} \displaystyle a &= -1, b=2, c=0 \\ \Delta &= b^2-4ac \\ &= 2^2 - 4 \times (-1) \times 0 \\ &= 4 \gt 0 \\ \end{align} \)

The graph cuts the $x$-axis twice and and the graph is above or below the $x$-axis depending on $x$ values.

This case is Not Negative Definite because $-x^2+2x$ is not always negative for all real $x$.

## Positive Definite

Quadratics which are positive for all real values of $x$ like Case 1: $y=x^2-2x+2$\( \begin{align} \displaystyle ax^2+bx+c &\gt 0 \\ a &\gt 0 \\ b^2 - 4ac &\lt 0 \\ \end{align} \)

## Negative Definite

Quadratics which are negative for all real values of $x$ like Case 4: $y=-x^2+2x-2$\( \begin{align} \displaystyle ax^2+bx+c &\lt 0 \\ a &\lt 0 \\ b^2 - 4ac &\lt 0 \\ \end{align} \)