10.5 Graphing quadratic equations  (Page 6/15)

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 $y=a{x}^2+bx+c$ :
  $\begin{array}{cccc}\hfill {\text{y}}\mathbf{\text{-intercept}}& & & \hfill {\text{x}}\mathbf{\text{-intercepts}}\\ \hfill \text{Let}x=0\text{and solve for}y.& & & \hfill \text{Let}y=0\text{and solve for}x.\end{array}$
• 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.

Practice makes perfect

Recognize the Graph of a Quadratic Equation in Two Variables

In the following exercises, graph:

In the following exercises, determine if the parabola opens up or down.

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.

Find the Intercepts of a Parabola

In the following exercises, find the x - and y -intercepts.

Graph Quadratic Equations in Two Variables

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.