To find the equation of a parabola given three points, we can assume the general form of the equation:

$y = ax^2 + bx + c$

We have three points: $(0, 2.9)$, $(27.5, 3.8)$, and $(3.75, 4.8)$. By substituting these coordinates into the equation, we can set up a system of three equations:

1. For the point $(0, 2.9)$, substitute $x = 0$ and $y = 2.9$: $2.9 = a(0)^2 + b(0) + c \quad \Rightarrow \quad c = 2.9$

2. For the point $(27.5, 3.8)$, substitute $x = 27.5$ and $y = 3.8$: $3.8 = a(27.5)^2 + b(27.5) + 2.9$ Simplifying this: $3.8 = a(756.25) + b(27.5) + 2.9$ $0.9 = 756.25a + 27.5b \quad \text{(Equation 1)}$

3. For the point $(3.75, 4.8)$, substitute $x = 3.75$ and $y = 4.8$: $4.8 = a(3.75)^2 + b(3.75) + 2.9$ Simplifying this: $4.8 = a(14.0625) + b(3.75) + 2.9$ $1.9 = 14.0625a + 3.75b \quad \text{(Equation 2)}$

Now, we solve the system of two equations:

1. $0.9 = 756.25a + 27.5b$
2. $1.9 = 14.0625a + 3.75b$

Let me solve this system of equations to find the values of $a$ and $b$. sympy import symbols, Eq, solve

Define variables

a, b = symbols('a b')

Equations

eq1 = Eq(756.25 * a + 27.5 * b, 0.9) eq2 = Eq(14.0625 * a + 3.75 * b, 1.9)

Solve the system of equations

solution = solve((eq1, eq2), (a, b)) solutionThe solution to the system of equations gives the values:

$a \approx -0.01996 \quad \text{and} \quad b \approx 0.5815$

Thus, the equation of the parabola is:

$y = -0.01996x^2 + 0.5815x + 2.9$

Would you like more details or have any questions about this process? Here are some related questions you might find useful:

1. How can I graph the parabola using its equation?
2. How would the equation change if the points were different?
3. What is the vertex form of a parabola, and how does it relate to this standard form?
4. How do I determine the direction of the parabola (upward or downward)?
5. How can I find the axis of symmetry for this parabola?

Tip: When solving systems of equations, using substitution or elimination methods can make the process much easier.