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Understanding Quadratic Equations through Graphing: Solutions and Zeros Explained

In this lesson, we review how to solve quadratic equations by graphing. We explore the concept of zeros—where the graph intersects the x-axis—and discuss how many solutions a quadratic can have. Quadratics can have two, one, or no solutions based on their graph behavior. We'll examine various graphs, identify the solutions (x-intercepts), and learn how to write the corresponding equations from these graphs. Use tables to find solutions and understand the relationships between graphing and quadratic equations more deeply.

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Understanding Quadratic Equations through Graphing: Solutions and Zeros Explained

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  1. Solving Quadratic Equations By Graphing

  2. Vocab review Do you remember from unit 1? Zeros – the place where the graph crosses the x-axis A.K.A – x-intercepts or Roots or solutions Zeros

  3. How many solutions does a Quadratic have? Sometimes is may have less than 2 solutions. Let’s see what that looks like. Two solutions because the highest exponent is 2

  4. How many solutions does a Quadratic have? 2 solutions The graph crosses the x-axis twice 1 solution The graph crosses the x-axis once No solution The graph never crosses the x-axis

  5. How many solutions does a Quadratic have? 2 solutions The graph crosses the x-axis twice 1 solution The graph crosses the x-axis once No solution The graph never crosses the x-axis

  6. Your turn! • How many solutions do each of these graphs have? 1. 2. 3. 1 solution 2 solutions 2 solutions

  7. Identifying the solutions from the graph X = -2, 2 X = -3 X = -4, 1

  8. You try! • Identify the solutions from the graph X = -2, 2 No solution X = 0

  9. Writing the equation from the graph. • Using the solutions from 2 slides ago X = -2, 2 X = -3 X = -4, 1 Y = (x + 3)2 Y = -(x + 4)(x – 1) Y = (x – 2)(x + 2)

  10. You try! Write the equation of each graph. 1. 2. 3. Y = (x + 5)(x – 3) Y = (x + 4)(x – 1) Y = - (x – 4)2

  11. Finding the solutions from a table Ex: 1 This table is of a Quadratic with 2 two solutions Remember: The solution is also called a zero or x-intercept. Meaning that the y-coordinate of the solution is 0 These are my solutions X = 5, 3

  12. Finding the sol’n from a table cont. Ex: 2 Find the solutions. X = -1 and 1

  13. Finding the sol’n from a table cont. Ex: 3 Find the solutions. This problem only has one solution X = 2 Ex: 4 This problem no solution

  14. Finding the sol’n from a table cont. Ex: 5 Find the solutions. This problem has two solution but only one is listed on the table X = 0 , 4

  15. Finding the solutions from a graph There are zeros between -5 and -4 and between 1 and 2

  16. Solutions from a table cont. Ex: 6 Find the interval where the zero is. The zeros are between 2 and 3 and between 5 and 6

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