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.
![## Problem Statement for Rational Function
### Task
Write an equation for a rational function with the following characteristics:
#### Given Characteristics:
1. **Vertical Asymptotes**:
\[
x = 6 \quad \text{and} \quad x = 1
\]
2. **x-intercepts**:
\[
x = -2 \quad \text{and} \quad x = 3
\]
3. **y-intercept**:
\[
y = 9
\]
### Solution Format:
You need to enter the equation of the rational function in the provided input box.
\[
y = \quad \text{(input box)}
\]
### Hints and Support:
- To get hints or help, you can watch a supporting video or post your questions on the forum.
**Buttons:**
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This problem involves creating an equation of a rational function that satisfies the specified conditions for vertical asymptotes, x-intercepts, and y-intercepts.
### Steps:
1. To include vertical asymptotes at \(x = 6\) and \(x = 1\), the denominator of the rational function will have factors \((x - 6)\) and \((x - 1)\).
2. To include x-intercepts at \(x = -2\) and \(x = 3\), the numerator of the rational function will have factors \((x + 2)\) and \((x - 3)\).
3. Determine the constant factor \(A\) by using the y-intercept information given \(y = 9\) when \(x = 0\).
\[ y = \frac{A(x + 2)(x - 3)}{(x - 6)(x - 1)} \]
To find \(A\):
When \(x = 0\), \( y = 9\):
\[ 9 = \frac{A(0 + 2)(0 - 3)}{(0 - 6)(0 - 1)} \]
Solve for \(A\):
\[ 9 = \frac{A(2)(-3)}{(-6)(-1)} \]
\[ 9 = \frac{-6A}{6} \]
\[ 9 = -A \]
\[ A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F633100ea-0428-4ac9-9895-e60fbf23e986%2F0196afed-0502-4141-a49a-c9445f5e7f6c%2Fv00odcq.png&w=3840&q=75)
Step by step
Solved in 2 steps with 2 images
