https://file.aiquickdraw.com/mathbot/file/1726864792840lge2ega1.jpg

From the image, you have a table with values for \( x \) and \( y \), where \( y \) is a function of \( x \). The points provided are:

\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 0 \\
-2 & 0 \\
-1 & -1 \\
0 & 0 \\
1 & 3 \\
2 & 8 \\
3 & 15 \\
\end{array}
\]

You're asked to find the second difference and the values of the quadratic function coefficients \( a \), \( b \), and \( c \) for the function \( y = ax^2 + bx + c \).

### Step 1: Finding the Second Difference
To find the second difference, calculate the differences between consecutive \( y \)-values.

#### First differences:
\[
\Delta_1 y = 
\begin{array}{c}
0 - 0 = 0 \\
-1 - 0 = -1 \\
0 - (-1) = 1 \\
3 - 0 = 3 \\
8 - 3 = 5 \\
15 - 8 = 7 \\
\end{array}
\]

Thus, the first differences are: \( 0, -1, 1, 3, 5, 7 \).

#### Second differences:
\[
\Delta_2 y = 
\begin{array}{c}
-1 - 0 = -1 \\
1 - (-1) = 2 \\
3 - 1 = 2 \\
5 - 3 = 2 \\
7 - 5 = 2 \\
\end{array}
\]

The second differences are constant at 2, confirming that this is a quadratic function. The second difference is 2.

### Step 2: Determining the Quadratic Function \( y = ax^2 + bx + c \)
The second difference being constant implies the function is quadratic. The coefficient \( a \) is half of the second difference, so:
\[
a = \frac{2}{2} = 1
\]

Now, substitute the known \( x \) and \( y \) values into the quadratic function \( y = ax^2 + bx + c \) to form a system of equations to find \( b \) and \( c \).

Using the points \( (-3, 0) \), \( (0, 0) \), and \( (1, 3) \):

1. For \( x = -3 \), \( y = 0 \):
\[
0 = 1(-3)^2 + b(-3) + c \quad \Rightarrow \quad 0 = 9 - 3b + c
\]
\[
9 - 3b + c = 0 \quad \Rightarrow \quad 3b - c = 9 \quad \text{(Equation 1)}
\]

2. For \( x = 0 \), \( y = 0 \):
\[
0 = 1(0)^2 + b(0) + c \quad \Rightarrow \quad 0 = c
\]
So, \( c = 0 \).

3. For \( x = 1 \), \( y = 3 \):
\[
3 = 1(1)^2 + b(1) + 0 \quad \Rightarrow \quad 3 = 1 + b
\]
\[
b = 2
\]

### Step 3: Final Function
Thus, the quadratic function is:
\[
y = x^2 + 2x
\]

### Final Answers:
- Second difference: 2
- \( a = 1 \)
- \( b = 2 \)
- \( c = 0 \)

Let me know if you'd like more details or if you have any questions! Here are five follow-up questions for practice:

1. How do you find the first difference of a sequence?
2. Why does a constant second difference imply a quadratic function?
3. What happens if the second difference is not constant?
4. How can you use other points to verify the values of \( a \), \( b \), and \( c \)?
5. How does changing \( a \) in the quadratic equation affect the graph?

**Tip:** The second difference method works only for quadratic sequences! If the second difference is not constant, the function is not quadratic.

For the quadratic function above, find the second difference and the values of the coefficients a, b, and c for the function y = ax^2 + bx + c.

Learn how to find the coefficients of a quadratic function y = ax^2 + bx + c using second differences. The problem involves calculating first and second differences, identifying the coefficient 'a', and solving for 'b' and 'c' using given points. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation