Graphing Quadratic Equations Summary

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 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.

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 \(x=0.\)