# 10.2: Quadratic Equations by Graphing

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Identify the number of solutions of quadratic equations.
- Solve quadratic equations by graphing.
- Find or approximate zeros of quadratic functions.
- Analyze quadratic functions using a graphing calculator.
- Solve real-world problems by graphing quadratic functions.

## Vocabulary

Terms introduced in this lesson:

- roots
- zeros
- distinct solutions
- double root
- no real solutions

## Teaching Strategies and Tips

Emphasize the connection between algebra and geometry:

- Finding the roots of a quadratic function (algebra) is equivalent to finding the \begin{align*}x-\end{align*}intercepts of a parabola (geometry).
- Have students use correct vocabulary: equations have
*roots*or*zeros*; graphs have \begin{align*}x-\end{align*}*intercepts*.

Use Example 4 to explore the graph of an equation using a graphing calculator.

- In graph mode, use the cursor to scroll over the \begin{align*}x-\end{align*}intercepts or vertex to find an approximate value of each. Approximations can be improved by zooming in.
- In graph mode, from the CALC menu, use ZERO and MAXIMUM (MINIMUM) to find an \begin{align*}x-\end{align*}intercept and vertex, respectively.
- Use the built-in table to find an \begin{align*}x-\end{align*}intercept (look for the row with \begin{align*}y = 0\end{align*}) or a vertex (scroll through the table values until the \begin{align*}y\end{align*} values change from increasing to decreasing; or vice-versa). By changing the table’s step size, the approximation can be made better.
- Emphasize approximating roots of an equation by reading a graph is an essential skill; real-world equations rarely factor over the integers.

In Example 5, encourage students to explore the equation on a graphing calculator using the WINDOW menu. By changing parameters XMIN, XMAX, YMIN, and YMAX, students learn to find an appropriate display for any graph.

Have students interpret their answers. In Example 5, the two roots indicate the two times when the arrow is on the ground.

General Tip: In *Review Questions* 1-12, have students check their answers with a graphing calculator for added practice.

## Error Troubleshooting

In *Review Question* 6, remind those students inputting the equation into a calculator to use proper syntax. The following are equivalent:

- \begin{align*}y_1 = (1/2)x^{\land}2-2x+3\end{align*}
- \begin{align*}y_1 = x^{\land}{2/2}-2x+3\end{align*}
- \begin{align*}y_1 = (x^{\land}2)/2 - 2x + 3\end{align*}