Math Problem Statement

Write the quadratic function that goes through the points (0,8) (2,18) (3,17)

Solution

To find a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c that passes through the points (0,8)(0, 8), (2,18)(2, 18), and (3,17)(3, 17), we can use the fact that each point will satisfy this function, giving us a system of equations for aa, bb, and cc.

Step 1: Set up equations

  1. Using the point (0,8)(0, 8): f(0)=a(0)2+b(0)+c=8c=8f(0) = a(0)^2 + b(0) + c = 8 \Rightarrow c = 8

  2. Using the point (2,18)(2, 18): f(2)=a(2)2+b(2)+c=18f(2) = a(2)^2 + b(2) + c = 18 Substituting c=8c = 8: 4a+2b+8=184a+2b=102a+b=54a + 2b + 8 = 18 \Rightarrow 4a + 2b = 10 \Rightarrow 2a + b = 5

  3. Using the point (3,17)(3, 17): f(3)=a(3)2+b(3)+c=17f(3) = a(3)^2 + b(3) + c = 17 Substituting c=8c = 8: 9a+3b+8=179a+3b=93a+b=39a + 3b + 8 = 17 \Rightarrow 9a + 3b = 9 \Rightarrow 3a + b = 3

Step 2: Solve the system of equations for aa and bb

We now have two equations:

  1. 2a+b=52a + b = 5
  2. 3a+b=33a + b = 3

Subtract the first equation from the second: (3a+b)(2a+b)=35(3a + b) - (2a + b) = 3 - 5 a=2a = -2

Substitute a=2a = -2 into the first equation: 2(2)+b=52(-2) + b = 5 4+b=5-4 + b = 5 b=9b = 9

Step 3: Write the quadratic function

We found that a=2a = -2, b=9b = 9, and c=8c = 8. So, the quadratic function is: f(x)=2x2+9x+8f(x) = -2x^2 + 9x + 8

Final Answer:

f(x)=2x2+9x+8f(x) = -2x^2 + 9x + 8

Would you like to go over the verification or any specific details?

  1. How can we verify that the function passes through each point?
  2. What other forms can a quadratic function take?
  3. How does changing aa, bb, or cc affect the shape of the graph?
  4. How do we find the vertex of this quadratic function?
  5. Can this quadratic function have real roots?

Tip: To confirm if a quadratic function fits specific points, plug each point into the function to see if the resulting equations hold true.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Systems of Linear Equations

Formulas

Quadratic function f(x) = ax^2 + bx + c
Substitution to solve systems of equations

Theorems

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Suitable Grade Level

Grades 10-12