## Key Concepts

• 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
1. Write the quadratic equation with $$y$$ on one side.
2. Determine whether the parabola opens upward or downward.
3. Find the axis of symmetry.
4. Find the vertex.
5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
6. Find the x-intercepts.
7. Graph the parabola.
• Minimum or Maximum Values of a Quadratic Equation
• The ycoordinate 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.

## 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 $$x=0.$$

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