Answer :
To find the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] given the conditions [tex]\( f(1) = -10 \)[/tex], [tex]\( f(-3) = 82 \)[/tex], and [tex]\( f(3) = 28 \)[/tex], we need to set up a system of equations using these points.
1. Use the point [tex]\( (1, -10) \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( f(x) = -10 \)[/tex] into the function:
[tex]\[
a(1)^2 + b(1) + c = -10
\][/tex]
Simplifying gives us the first equation:
[tex]\[
a + b + c = -10
\][/tex]
2. Use the point [tex]\( (-3, 82) \)[/tex]:
Substitute [tex]\( x = -3 \)[/tex] and [tex]\( f(x) = 82 \)[/tex] into the function:
[tex]\[
a(-3)^2 + b(-3) + c = 82
\][/tex]
Simplifying gives us the second equation:
[tex]\[
9a - 3b + c = 82
\][/tex]
3. Use the point [tex]\( (3, 28) \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( f(x) = 28 \)[/tex] into the function:
[tex]\[
a(3)^2 + b(3) + c = 28
\][/tex]
Simplifying gives us the third equation:
[tex]\[
9a + 3b + c = 28
\][/tex]
Now, we have a system of three equations:
[tex]\[
\begin{align*}
1) & \quad a + b + c = -10 \\
2) & \quad 9a - 3b + c = 82 \\
3) & \quad 9a + 3b + c = 28
\end{align*}
\][/tex]
To solve this system, we can use elimination or substitution. Here’s a simplified method:
- Subtract equation (1) from equations (2) and (3):
From equation (2) minus equation (1):
[tex]\[
(9a - 3b + c) - (a + b + c) = 82 - (-10)
\][/tex]
Simplifying:
[tex]\[
8a - 4b = 92 \quad \Rightarrow \quad 2a - b = 23
\][/tex]
From equation (3) minus equation (1):
[tex]\[
(9a + 3b + c) - (a + b + c) = 28 - (-10)
\][/tex]
Simplifying:
[tex]\[
8a + 2b = 38 \quad \Rightarrow \quad 4a + b = 19
\][/tex]
- Solve the two new equations:
[tex]\[
2a - b = 23
\][/tex]
[tex]\[
4a + b = 19
\][/tex]
Adding these two equations:
[tex]\[
6a = 42 \quad \Rightarrow \quad a = 7
\][/tex]
Substitute [tex]\( a = 7 \)[/tex] into [tex]\( 2a - b = 23 \)[/tex]:
[tex]\[
2(7) - b = 23 \quad \Rightarrow \quad 14 - b = 23 \quad \Rightarrow \quad b = -9
\][/tex]
Substitute [tex]\( a = 7 \)[/tex] and [tex]\( b = -9 \)[/tex] into equation 1:
[tex]\[
7 - 9 + c = -10 \quad \Rightarrow \quad -2 + c = -10 \quad \Rightarrow \quad c = -8
\][/tex]
Thus, the quadratic function is [tex]\( f(x) = 7x^2 - 9x - 8 \)[/tex].
1. Use the point [tex]\( (1, -10) \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( f(x) = -10 \)[/tex] into the function:
[tex]\[
a(1)^2 + b(1) + c = -10
\][/tex]
Simplifying gives us the first equation:
[tex]\[
a + b + c = -10
\][/tex]
2. Use the point [tex]\( (-3, 82) \)[/tex]:
Substitute [tex]\( x = -3 \)[/tex] and [tex]\( f(x) = 82 \)[/tex] into the function:
[tex]\[
a(-3)^2 + b(-3) + c = 82
\][/tex]
Simplifying gives us the second equation:
[tex]\[
9a - 3b + c = 82
\][/tex]
3. Use the point [tex]\( (3, 28) \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( f(x) = 28 \)[/tex] into the function:
[tex]\[
a(3)^2 + b(3) + c = 28
\][/tex]
Simplifying gives us the third equation:
[tex]\[
9a + 3b + c = 28
\][/tex]
Now, we have a system of three equations:
[tex]\[
\begin{align*}
1) & \quad a + b + c = -10 \\
2) & \quad 9a - 3b + c = 82 \\
3) & \quad 9a + 3b + c = 28
\end{align*}
\][/tex]
To solve this system, we can use elimination or substitution. Here’s a simplified method:
- Subtract equation (1) from equations (2) and (3):
From equation (2) minus equation (1):
[tex]\[
(9a - 3b + c) - (a + b + c) = 82 - (-10)
\][/tex]
Simplifying:
[tex]\[
8a - 4b = 92 \quad \Rightarrow \quad 2a - b = 23
\][/tex]
From equation (3) minus equation (1):
[tex]\[
(9a + 3b + c) - (a + b + c) = 28 - (-10)
\][/tex]
Simplifying:
[tex]\[
8a + 2b = 38 \quad \Rightarrow \quad 4a + b = 19
\][/tex]
- Solve the two new equations:
[tex]\[
2a - b = 23
\][/tex]
[tex]\[
4a + b = 19
\][/tex]
Adding these two equations:
[tex]\[
6a = 42 \quad \Rightarrow \quad a = 7
\][/tex]
Substitute [tex]\( a = 7 \)[/tex] into [tex]\( 2a - b = 23 \)[/tex]:
[tex]\[
2(7) - b = 23 \quad \Rightarrow \quad 14 - b = 23 \quad \Rightarrow \quad b = -9
\][/tex]
Substitute [tex]\( a = 7 \)[/tex] and [tex]\( b = -9 \)[/tex] into equation 1:
[tex]\[
7 - 9 + c = -10 \quad \Rightarrow \quad -2 + c = -10 \quad \Rightarrow \quad c = -8
\][/tex]
Thus, the quadratic function is [tex]\( f(x) = 7x^2 - 9x - 8 \)[/tex].
