## Key Concepts

Contents

**The graph of every quadratic equation is a parabola.****Parabola Orientation**For the quadratic equation \(y=a{x}^{2}+bx+c\), if- \(a>0\), the parabola opens upward.
- \(a<0\), the parabola opens downward.

**Axis of Symmetry and Vertex of a Parabola**For a parabola with equation \(y=a{x}^{2}+bx+c\):- The axis of symmetry of a parabola is the line \(x=-\frac{b}{2a}\).
- The vertex is on the axis of symmetry, so its
*x*-coordinate is \(-\frac{b}{2a}\). - To find the
*y*-coordinate of the vertex we substitute \(x=-\frac{b}{2a}\) into the quadratic equation.

**Find the Intercepts of a Parabola**To find the intercepts of a parabola with equation \(y=a{x}^{2}+bx+c\):\(\begin{array}{cccc}\hfill {\text{y}}\mathbf{\text{-intercept}}& & & \hfill {\text{x}}\mathbf{\text{-intercepts}}\\ \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}x=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}y.& & & \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}y=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}x.\end{array}\)

**To Graph a Quadratic Equation in Two Variables**- Write the quadratic equation with \(y\) on one side.
- Determine whether the parabola opens upward or downward.
- Find the axis of symmetry.
- Find the vertex.
- Find the
*y*-intercept. Find the point symmetric to the*y*-intercept across the axis of symmetry. - Find the
*x*-intercepts. - Graph the parabola.

**Minimum or Maximum Values of a Quadratic Equation**- The
*y*–**coordinate of the vertex**of the graph of a quadratic equation is the **minimum**value of the quadratic equation if the parabola opens upward.**maximum**value of the quadratic equation if the parabola opens downward.

- The

## Glossary

### axis of symmetry

The axis of symmetry is the vertical line passing through the middle of the parabola \(y=a{x}^{2}+bx+c.\)

### parabola

The graph of a quadratic equation in two variables is a parabola.

### quadratic equation in two variables

A quadratic equation in two variables, where \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are real numbers and \(a\ne 0\) is an equation of the form \(y=a{x}^{2}+bx+c.\)

### vertex

The point on the parabola that is on the axis of symmetry is called the *vertex* of the parabola; it is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards.

*x*-intercepts of a parabola

The *x*-intercepts are the points on the parabola where \(y=0.\)

*y*-intercept of a parabola

The *y*-intercept is the point on the parabola where \(x=0.\)