Question

create a equation from the 3 points (0,10),(1,10),(2,0)

Answer 1

To create an equation from the three points \((0,10)\), \((1,10)\), and \((2,0)\), we can start by identifying the type of function that fits these points. Since we have three points, a quadratic function of the form \(y = ax^2 + bx + c\) could fit the data.

Let's denote the points as:
1. \((0, 10)\)
2. \((1, 10)\)
3. \((2, 0)\)

We can create a system of equations using these points.

1. From the point \((0, 10)\):
\[
c = 10
\]

2. From the point \((1, 10)\):
\[
a(1)^2 + b(1) + c = 10 \implies a + b + 10 = 10 \implies a + b = 0 \quad \text{(Equation 1)}
\]

3. From the point \((2, 0)\):
\[
a(2)^2 + b(2) + c = 0 \implies 4a + 2b + 10 = 0 \implies 4a + 2b = -10 \quad \text{(Equation 2)}
\]

Now we solve the system of equations.

From Equation 1:
\[
b = -a
\]

Substituting \(b = -a\) into Equation 2:
\[
4a + 2(-a) = -10 \implies 4a - 2a = -10 \implies 2a = -10 \implies a = -5
\]

Now substituting \(a = -5\) back into Equation 1 to find \(b\):
\[
-5 + b = 0 \implies b = 5
\]

Now we have \(a\), \(b\), and \(c\):
\[
a = -5, \quad b = 5, \quad c = 10
\]

Thus, the quadratic equation that fits the points \((0,10)\), \((1,10)\), and \((2,0)\) is:
\[
y = -5x^2 + 5x + 10
\]

This is the equation based on the given points.