Answer :
Quadratic function ( in standard form ):
y = a x² + b x + c
x = 0:
635 = 0 + 0 + c
c = 635
x = 1:
644 = a + b + 635
a + b = 9
x = 2:
719 = 4 a + 2 b + 635
4 a + 2 b = 84
and now we have a system:
a + b = 9 / * ( - 2 )
4 a + 2 b = 84
-------------------------
- 2 a + 2 b = - 18
+
4 a + 2 b = 84
-------------------------
2 a = 66
a = 33; b = - 24;
Function is:
f ( x ) = 33 x² - 24 x + 635
f ( 8 ) = 33 * 64 - 24 * 8 + 635 = 2,112 - 192 + 635 = 2,555
Answer: The number of waterfowl on week 8 would be 2,555.
y = a x² + b x + c
x = 0:
635 = 0 + 0 + c
c = 635
x = 1:
644 = a + b + 635
a + b = 9
x = 2:
719 = 4 a + 2 b + 635
4 a + 2 b = 84
and now we have a system:
a + b = 9 / * ( - 2 )
4 a + 2 b = 84
-------------------------
- 2 a + 2 b = - 18
+
4 a + 2 b = 84
-------------------------
2 a = 66
a = 33; b = - 24;
Function is:
f ( x ) = 33 x² - 24 x + 635
f ( 8 ) = 33 * 64 - 24 * 8 + 635 = 2,112 - 192 + 635 = 2,555
Answer: The number of waterfowl on week 8 would be 2,555.
The number of waterfowl on 8 week would be [tex]\boxed{2555}[/tex].
Further explanation:
The general form of the quadratic equation is given by,
[tex]\boxed{a{x^2} + bx + c}[/tex]
Here, a is the coefficient of [tex]{x^2}[/tex], b is the coefficient of [tex]x[/tex] and [tex]c[/tex] is the constant term.
Given:
The population of the lake in 0, 1, 2, 3, 4, 5 and 6 week is 635, 644, 719, 860, 1067, 1340 and 1769 respectively.
Calculation:
Consider [tex]x[/tex] denote the number of weeks.
Quadratic function that models the data as the function of [tex]x[/tex] can be expressed as,
[tex]\boxed{N=a{x^2}+bx +c}[/tex]
Here, [tex]N[/tex] denotes the population in [tex]{n^{th}}[/tex] week.
The population of lake in the beginning is 635. Therefore, value of [tex]x[/tex] is zero and value of [tex]N[/tex] is 635.
Substitute [tex]0[/tex] for [tex]x[/tex] and [tex]635[/tex] for [tex]N[/tex] in quadratic equation [tex]N = a{x^2} + bx + c[/tex].
[tex]\begin{aligned} 635&=a{\left(0\right)^2}+b\left( 0 \right)+c\\ 635&=0+0+c\\ 635&=c\\\end{aligned}[/tex]
The population of lake after one week is [tex]644[/tex]. Therefore, value of [tex]x[/tex] is one and value of [tex]N[/tex] is [tex]644[/tex].
Substitute [tex]1[/tex] for [tex]x[/tex] and [tex]644[/tex] for [tex]N[/tex] in quadratic equation [tex]N = a{x^2} + bx + c[/tex].
[tex]\begin{aligned} 644 &= a{\left( 1 \right)^2} + b\left( 1 \right) + c \\ 644 &= a + b + c \\ 644 &= a + b + 635 \\ 644 - 635 &= a + b \\ 9 &= a + b \\\end{aligned}[/tex]
The population of lake after two week is [tex]719[/tex]. Therefore, value of [tex]x[/tex] is two and value of [tex]N[/tex] is [tex]719[/tex].
Substitute 2 for [tex]x[/tex] and 719 for [tex]N[/tex] in quadratic equation [tex]N = a{x^2} + bx + c[/tex].
[tex]\begin{aligned} 719 &= a{\left( 2 \right)^2} + b\left( 2 \right) + c \\ 719 &= 4a + 2b + 635 \\ 719 - 635 &= 4a + 2b \\ 84 &= 2\left( {2a + b} \right) \\ \frac{{84}}{2} &= \left( {2a + b} \right) \\ 42 &= \left( {2a + b} \right) \\ \end{aligned}[/tex]
Solve the equation [tex]a + b = 9[/tex] to obtain the value [tex]a[/tex].
[tex]a = 9 - b[/tex]
Substitute [tex]9 - b[/tex] for in equation [tex]42 = 2a + b[/tex].
[tex]\begin{aligned} 42 &= 2\left( {9 - b} \right) + b \\42 &= 18 - 2b + b \\ 42 - 18 &=- b \\24 &=- b\\- 24 &= b \\ \end{aligned}[/tex]
Substitute [tex]-24[/tex] for [tex]b[/tex] in equation [tex]a = 9 - b[/tex] to obtain the value of a.
[tex]\begin{aligned}a&= 9-\left( {-24} \right) \\&= 9+24\\&= 33 \\\end{aligned}[/tex]
To obtain the quadratic function substitute 33 for a, -24 for b and 635 for c in equation [tex]N = a{x^2} + bx + c[/tex].
[tex]N = 33{x^2} - 24x + 635[/tex]
Substitute 8 for [tex]x[/tex] in above equation to obtain the number of waterfowl at the lake after week [tex]8[/tex].
[tex]\begin{aligned} N &= 33{\left( 8 \right)^2} - 24\left( 8 \right) + 635 \\ &=33\left( {64} \right) - 192 + 635 \\ &= 2112 - 192 + 635 \\ &= 2555 \\ \end{aligned}[/tex]
The number of waterfowl on [tex]8[/tex] week would be [tex]\boxed{2555}[/tex].
Learn more:
1. Learn more about product of binomial and trinomial https://brainly.com/question/1394854
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Quadratic equation
Keywords: quadratic equation, polynomial, square root, real numbers, coefficients, constant term, general form of quadratic equation, waterfowl, lake, population, migrating, a biologist, model, estimate, week, recounted, function.