Find the formula for a quadratic function that satisfies the given conditions. Vertex (4,2) passing through the poin: (6,14) f(x) = ] (Simplify your answer.)
Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
![The task is to find the formula for a quadratic function that satisfies the given conditions: the vertex of the function is at the point (4, 2), and it passes through the point (6, 14).
### Problem Details:
- **Vertex**: (4, 2)
- **Point**: (6, 14)
The quadratic function can be expressed in vertex form as:
\[ f(x) = a(x - h)^2 + k \]
Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a coefficient that determines the direction and width of the parabola.
### Steps:
1. **Substitute the Vertex into the Formula**:
\[ f(x) = a(x - 4)^2 + 2 \]
2. **Use the Given Point to Find \( a \)**:
Substitute point (6, 14) into the equation:
\[ 14 = a(6 - 4)^2 + 2 \]
3. **Solve for \( a \)**:
\[ 14 = a(2)^2 + 2 \]
\[ 14 = 4a + 2 \]
\[ 12 = 4a \]
\[ a = 3 \]
4. **Write the Final Quadratic Function**:
\[ f(x) = 3(x - 4)^2 + 2 \]
The answer box for input is provided below the question with options to simplify or clear the answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9363fbf8-1ca8-4725-9136-263a3c8a662e%2F064b629e-da0c-4852-9b9a-790e546e0659%2F4h8j6w_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
