Plot the ordered pairs (1,3), (3,-5), (-1,-5) on the coordinate plane. Which of the following functions goes through the three given points?

To determine the function that goes through the three given points $(1,3)$, $(3,-5)$, and $(-1,-5)$, we will follow these steps:

1. Break Down the Problem

• Identify the coordinates of the three points.
• Check if the points are collinear or if they can form a quadratic function.

2. Relevant Concepts

• For three points, we can check if they form a linear function $y = mx + b$ or a quadratic function of the form $y = ax^2 + bx + c$.
• Use the concept of slopes to check for linearity.

3. Analysis and Detail

1. Checking if Points are Collinear:

   • Calculate the slope between the points: $m_{(1,3) \text{ and } (3,-5)} = \frac{-5 - 3}{3 - 1} = \frac{-8}{2} = -4$ $m_{(3,-5) \text{ and } (-1,-5)} = \frac{-5 - (-5)}{-1 - 3} = \frac{0}{-4} = 0$
     • Since $m_{(1,3) \text{ and } (3,-5)} \neq m_{(3,-5) \text{ and } (-1,-5)}$, the points are not collinear.

2. Finding a Quadratic Function:

   • Assume the form $y = ax^2 + bx + c$.
   • Substitute the points into the equation: $3 = a(1^2) + b(1) + c \quad (1)$ $-5 = a(3^2) + b(3) + c \quad (2)$ $-5 = a(-1^2) + b(-1) + c \quad (3)$

3. Set up the System of Equations:

   • From equation (1): $a + b + c = 3$
   • From equation (2): $9a + 3b + c = -5$
   • From equation (3): $a - b + c = -5$

4. Solving the System of Equations

• Subtract equation (1) from equation (2): $9a + 3b + c - (a + b + c) = -5 - 3 \\ 8a + 2b = -8 \Rightarrow 4a + b = -4 \quad (4)$

• Subtract equation (1) from equation (3): $a - b + c - (a + b + c) = -5 - 3 \\ -2b = -8 \Rightarrow b = 4 \quad (5)$

• Substitute $b = 4$ into equation (4): $4a + 4 = -4 \Rightarrow 4a = -8 \Rightarrow a = -2$

• Substitute $a$ and $b$ back into equation (1): $-2 + 4 + c = 3 \Rightarrow c = 1$

Final Answer

The quadratic function that goes through the points $(1,3)$, $(3,-5)$, and $(-1,-5)$ is: $y = -2x^2 + 4x + 1$