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Find the equation of the parabola by using three points (4,0), (2,8), and (6,0).

Find the equation of the parabola by using three points (4,0), (2,8), and (6,0).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given the graph of the parabola

### Understanding Parabolas: Finding the Equation from Points #### Graph and Function Analysis **Graph Description:** - The graph displays a parabola opening upwards. - The vertex of the parabola appears to be at the point (2, -4). - The parabola passes through points (4,0), (2,8), and (6,0) as given. #### Task: Determine the Equation To find the equation of the parabola using these points: **Step 1: Formulate the System of Equations** Set up equations using the standard form of a quadratic equation \( y = ax^2 + bx + c \). Use the points (4,0), (2,8), and (6,0) and submit these into the equation: 1. \( a(4)^2 + b(4) + c = 0 \) 2. \( a(2)^2 + b(2) + c = 8 \) 3. \( a(6)^2 + b(6) + c = 0 \) Fill in the blanks to form these equations. **Step 2: Solve the Equation** Find the coefficients \( a \), \( b \), and \( c \) to complete the equation: \( y = ax^2 + bx + c \) Use the provided grid to substitute and find the solution. --- This structured method will help students learn how to derive quadratic equations from geometric representations on graphs effectively.
Transcribed Image Text:### Understanding Parabolas: Finding the Equation from Points #### Graph and Function Analysis **Graph Description:** - The graph displays a parabola opening upwards. - The vertex of the parabola appears to be at the point (2, -4). - The parabola passes through points (4,0), (2,8), and (6,0) as given. #### Task: Determine the Equation To find the equation of the parabola using these points: **Step 1: Formulate the System of Equations** Set up equations using the standard form of a quadratic equation \( y = ax^2 + bx + c \). Use the points (4,0), (2,8), and (6,0) and submit these into the equation: 1. \( a(4)^2 + b(4) + c = 0 \) 2. \( a(2)^2 + b(2) + c = 8 \) 3. \( a(6)^2 + b(6) + c = 0 \) Fill in the blanks to form these equations. **Step 2: Solve the Equation** Find the coefficients \( a \), \( b \), and \( c \) to complete the equation: \( y = ax^2 + bx + c \) Use the provided grid to substitute and find the solution. --- This structured method will help students learn how to derive quadratic equations from geometric representations on graphs effectively.
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