2-7: Solve Quadratic Equations by Graphing

By Oscar Castanos

​📅 Integrated Math II – Objective 2-7

Objective: 🎯 I can solve quadratic equations by graphing and relate solutions to the x-intercepts.

Sub-Objective: 🧩 I can identify the x-intercepts (zeros, roots, or solutions) from a graph and connect them to the equation’s factors or vertex form.

Warm-Up:
1️⃣ Review – Write an equation of a quadratic with vertex
(2, –3) and point (0, 5).
2️⃣ Lead-In – Predict where this graph will cross the x-axis. How do you know?

Today’s Plan:
- Warm-Up & Review - Notes: Connecting Equations ↔ Graphs - Mild → Medium - Practice
- Think-Pair-Share + Error Analysis
- Check Your Understanding (CYU)

​Warm-Up:
1️⃣ Review: Write an equation of a quadratic with vertex (2, –3) and point (0, 5).

2️⃣ Lead-In: Predict where this graph will cross the x-axis. How do you know?

​Concept Overview

Quadratic equations can be written in different forms—but each reveals something about the graph. For Example:

​Big Idea:
The solutions to y = 0 are the x-values where the graph crosses the x-axis. Those are your x-intercepts, zeros, or roots.

​Mild (Teacher Model)
Example 1 – Identifying Solutions from a Graph

Find the solutions (x-intercepts, zeros, or roots) of y = (x – 2)(x + 4) by graphing and algebraiclly.

​Mild (Student Try)
Try It – Find Solutions from a Graph

For each quadratic, determine the x-intercepts and write the solutions to y = 0 by graphing and algebraiclly.

a. y = (x + 1)(x – 3)
b. y = –(x – 4)(x – 2)

​Think-Pair-Share #1: Write your responses as a pair into your notes.
1. Partner A: names the intercepts.
2. Partner B: verifies by plugging into the equation.
3. Together: Explain what “solving by graphing” means in your own words.

​Medium (Teacher Model)
Example 2 – Solving When Given Vertex Form

Find the solutions for y = (x – 3)² – 9 graphically and algebraiclly.

​Medium (Student Try)
Try It – Vertex Form to Intercepts

Find the x-intercepts (solutions) graphically and algebraiclly for:

a. y = (x + 2)² – 4

b. y = –(x – 1)² + 9

​Error Analysis:

Common Error: A student says the solutions to
y = x² + 4 are x = ±2.

1. Graph the equation.

2. Looks for solutions (x-intercepts).

3. Write what the error is/are and your conclusion on the possible solutions.

​Think-Pair-Share (Compare Forms) Here are 3 forms of the same quadratic function:
1. y = x² – 6x + 8
2. y = (x – 2)(x – 4)
3. y = (x – 3)² – 1

Discuss:
Which form makes the x-intercepts easiest to find?

Which form shows the vertex clearest?

How can you move between these forms?

​CYU (Check Your Understanding)

Mild:
Identify the x-intercepts and solutions of y = (x – 1)(x + 5).

Medium:
Solve by graphing y = (x – 4)² – 9 and state the solutions.

Spicy (Application):
A ball is thrown from a height of 2 m. Its height after t seconds is
h = -5t²+10t+2 When does it hit the ground?