7, 195,334

8, 202,588

9, 221,327

10, 241,316

Can you find a cubic graph and equation that goes through these points please

To find a cubic equation that goes through the points:

- (7, 195,334)
- (8, 202,588)
- (9, 221,327)
- (10, 241,316)

A cubic equation is generally expressed as:

\[
y = ax^3 + bx^2 + cx + d
\]

We need to find the coefficients \(a\), \(b\), \(c\), and \(d\) by setting up a system of equations based on the given points.

For each point \((x, y)\), the equation becomes:

1. \( 195334 = a(7^3) + b(7^2) + c(7) + d \)
2. \( 202588 = a(8^3) + b(8^2) + c(8) + d \)
3. \( 221327 = a(9^3) + b(9^2) + c(9) + d \)
4. \( 241316 = a(10^3) + b(10^2) + c(10) + d \)

This gives us four linear equations, which we can solve for \(a\), \(b\), \(c\), and \(d\).

Let me calculate the values of \(a\), \(b\), \(c\), and \(d\).The cubic equation that goes through the points you provided is:

\[
y = -\frac{10235}{6}x^3 + \frac{93365}{2}x^2 - \frac{1214093}{3}x + 1325876
\]

Simplifying the coefficients into decimals for clarity:

\[
y = -1705.83x^3 + 46682.5x^2 - 404697.67x + 1325876
\]

This is the cubic equation that fits the given points.

Would you like more details on the process or have any questions? Here are a few related questions for further exploration:

1. What is the predicted value for \(x = 11\) using this equation?
2. How can we verify that the equation passes exactly through the given points?
3. What are the roots of this cubic equation, if any?
4. How can this equation be used to predict values for other \(x\)-coordinates?
5. How would the graph of this cubic equation look in comparison to a quadratic or linear graph?

**Tip**: You can use a system of equations solver or matrix methods to solve similar problems for higher degree polynomials.

Can you find a cubic graph and equation that goes through these points: (7, 195,334), (8, 202,588), (9, 221,327), (10, 241,316)?

Discover how to find a cubic equation that passes through the points (7, 195,334), (8, 202,588), (9, 221,327), and (10, 241,316). Learn to solve a system of equations to find the coefficients of a cubic function. This guide is ideal for students in Grades 10-12 studying polynomial equations and algebraic functions.

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