The quadratic function is [tex]f(x) =3x^2 -2x +5[/tex]
The table entries are given as:
x : -2, -1, 0, 1, 2.
f(x) : 21, 10, 5, 6, 13
A quadratic function is represented as:
[tex]f(x) = ax^2 + bx + c[/tex]
When x = 0, we have:
[tex]f(0) = a(0)^2 + b(0) + c[/tex]
[tex]f(0) = c[/tex]
From the table,
[tex]f(0) =5[/tex]
So, we have:
[tex]c = 5[/tex]
The quadratic function becomes
[tex]f(x) = ax^2 + bx + 5[/tex]
Also from the function, we have:
[tex]f(-1) = 10[/tex]
So, the function becomes
[tex]a(-1)^2 + b(-1) + 5 = 10[/tex]
[tex]a -b + 5 = 10[/tex]
Subtract 5 from both sides
[tex]a -b = 5[/tex]
Also from the function, we have:
[tex]f(1) = 6[/tex]
So, the function becomes
[tex]a(1)^2 + b(1) + 5 = 6[/tex]
[tex]a + b + 5 = 6[/tex]
Subtract 5 from both sides
[tex]a + b = 1[/tex]
Make a, the subject
[tex]a = 1 - b[/tex]
Substitute 1 - b for a in [tex]a -b = 5[/tex]
[tex](1 - b) - b= 5[/tex]
[tex]1 - b - b= 5[/tex]
[tex]1 - 2b= 5[/tex]
Subtract 1 from both sides
[tex]- 2b= 4[/tex]
Divide both sides by -2
[tex]b= -2[/tex]
Substitute -2 for b in [tex]a = 1 - b[/tex]
[tex]a=1+2[/tex]
[tex]a=3[/tex]
So, the function becomes
[tex]f(x) =3x^2 -2x +5[/tex]
Hence, the quadratic function is [tex]f(x) =3x^2 -2x +5[/tex]
Read more about quadratic functions at:
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