https://file.aiquickdraw.com/mathbot/file/1726188615818p3osr6md.jpg

The image asks to find a quadratic function \( y = ax^2 + bx + c \) that passes through the points \( (0,7), (2,13), \) and \( (4,3) \).

To solve this, we can plug the points into the quadratic equation to form a system of equations:

1. For \( (0, 7) \):  
   \( 7 = a(0)^2 + b(0) + c \)  
   This simplifies to:  
   \( c = 7 \)

2. For \( (2, 13) \):  
   \( 13 = a(2)^2 + b(2) + c \)  
   Plugging in \( c = 7 \):  
   \( 13 = 4a + 2b + 7 \)  
   Simplifying:  
   \( 4a + 2b = 6 \)  
   (Equation 1)

3. For \( (4, 3) \):  
   \( 3 = a(4)^2 + b(4) + c \)  
   Plugging in \( c = 7 \):  
   \( 3 = 16a + 4b + 7 \)  
   Simplifying:  
   \( 16a + 4b = -4 \)  
   (Equation 2)

Now, solve the system of equations:

**From Equation 1:**  
\( 4a + 2b = 6 \)  
Dividing by 2:  
\( 2a + b = 3 \)  
(Equation 3)

**From Equation 2:**  
\( 16a + 4b = -4 \)  
Dividing by 4:  
\( 4a + b = -1 \)  
(Equation 4)

Now, subtract Equation 3 from Equation 4:
\[
(4a + b) - (2a + b) = -1 - 3
\]
This simplifies to:
\[
2a = -4 \quad \Rightarrow \quad a = -2
\]

Substitute \( a = -2 \) into Equation 3:
\[
2(-2) + b = 3 \quad \Rightarrow \quad -4 + b = 3 \quad \Rightarrow \quad b = 7
\]

Now, we have \( a = -2 \), \( b = 7 \), and \( c = 7 \).

Thus, the quadratic equation is:
\[
y = -2x^2 + 7x + 7
\]

Let me know if you need any further clarification or details! Here are a few related questions:

1. How do you derive the quadratic equation from three points?
2. How does solving a system of equations work?
3. Can a quadratic function have more than two solutions?
4. How would you graph the quadratic function \( y = -2x^2 + 7x + 7 \)?
5. What are the properties of a parabola based on the quadratic equation?

**Tip:** The sign of the \( a \)-term in a quadratic equation determines whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

Find a quadratic function that includes the set of values: (0,7), (2,13), (4,3).

Learn how to find the quadratic function that passes through the points (0,7), (2,13), and (4,3). This solution covers the step-by-step process of solving for coefficients a, b, and c by forming and solving a system of equations. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation