For The Given Equation, Find The Values Of { A, B$}$, And { C$}$, Determine The Direction In Which The Parabola Opens, And Determine The { Y$}$-intercept. Decide Which Table Best Illustrates These Values For The

Mar 13, 2025 by ADMIN 212 views

**Solving for the Values of a, b, and c in a Quadratic Equation**

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is often written in the form of ${ax^2 + bx + c = 0}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${x}$ is the variable. In this article, we will focus on finding the values of ${a}$ , ${b}$ , and ${c}$ in a given quadratic equation, determining the direction in which the parabola opens, and finding the ${y}$ -intercept.

The Importance of a, b, and c in a Quadratic Equation

The values of ${a}$ , ${b}$ , and ${c}$ play a crucial role in determining the shape and position of the parabola. The value of ${a}$ determines the direction in which the parabola opens, while the values of ${b}$ and ${c}$ determine the position of the vertex and the ${y}$ -intercept.

Determining the Direction of the Parabola

The value of ${a}$ determines the direction in which the parabola opens. If ${a > 0}$ , the parabola opens upwards, while if ${a < 0}$ , the parabola opens downwards.

Determining the y-Intercept

The ${y}$ -intercept is the point at which the parabola intersects the ${y}$ -axis. It can be found by substituting ${x = 0}$ into the equation and solving for ${y}$ .

Finding the Values of a, b, and c

To find the values of ${a}$ , ${b}$ , and ${c}$ , we need to examine the given equation and identify the coefficients of the terms. The coefficient of the ${x^2}$ term is ${a}$ , the coefficient of the ${x}$ term is ${b}$ , and the constant term is ${c}$ .

Example: Solving for a, b, and c in a Quadratic Equation

Let's consider the quadratic equation ${x^2 + 5x + 6 = 0}$ . To find the values of ${a}$ , ${b}$ , and ${c}$ , we can identify the coefficients of the terms:

The coefficient of the ${x^2}$ term is ${a = 1}$ .
The coefficient of the ${x}$ term is ${b = 5}$ .
The constant term is ${c = 6}$ .

Determining the Direction of the Parabola

Since ${a = 1 > 0}$ , the parabola opens upwards.

Determining the y-Intercept

To find the ${y}$ -intercept, we can substitute ${x = 0}$ into the equation and solve for ${y}$ :

${y = (0)^2 + 5(0) + 6 = 6}$

Therefore, the ${y}$ -intercept is ${(0, 6)}$ .

Which Table Best Illustrates These Values?

To determine which table best illustrates the values of ${a}$ , ${b}$ , and ${c}$ , we need to examine the given tables and identify the one that accurately represents the values.

Table 1	Table 2	Table 3
a = 1, b = 5, c = 6	a = 1, b = 3, c = 2	a = 2, b = 4, c = 1

Based on the values of ${a}$ , ${b}$ , and ${c}$ , we can see that Table 1 accurately represents the values of the given quadratic equation.

Conclusion

In conclusion, finding the values of ${a}$ , ${b}$ , and ${c}$ in a quadratic equation is crucial in determining the direction in which the parabola opens and the ${y}$ -intercept. By identifying the coefficients of the terms, we can determine the values of ${a}$ , ${b}$ , and ${c}$ and use them to find the direction of the parabola and the ${y}$ -intercept.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is often written in the form of ${ax^2 + bx + c = 0}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${x}$ is the variable.

Q: How do I determine the direction of the parabola?

A: The value of ${a}$ determines the direction in which the parabola opens. If ${a > 0}$ , the parabola opens upwards, while if ${a < 0}$ , the parabola opens downwards.

Q: How do I find the y-intercept?

A: To find the ${y}$ -intercept, you can substitute ${x = 0}$ into the equation and solve for ${y}$ .

Q: What is the significance of the values of a, b, and c?

A: The values of ${a}$ , ${b}$ , and ${c}$ play a crucial role in determining the shape and position of the parabola. The value of ${a}$ determines the direction in which the parabola opens, while the values of ${b}$ and ${c}$ determine the position of the vertex and the ${y}$ -intercept.

Q: How do I solve for the values of a, b, and c?

A: To find the values of ${a}$ , ${b}$ , and ${c}$ , you need to examine the given equation and identify the coefficients of the terms. The coefficient of the ${x^2}$ term is ${a}$ , the coefficient of the ${x}$ term is ${b}$ , and the constant term is ${c}$ .

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a parabolic shape, while a linear equation has a straight line shape.

Q: Can a quadratic equation have more than one solution?

A: Yes, a quadratic equation can have more than one solution. In fact, a quadratic equation can have two solutions, one real and one complex, or two complex solutions.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the following steps:

Determine the direction of the parabola by examining the value of ${a}$ .
Find the ${y}$ -intercept by substituting ${x = 0}$ into the equation and solving for ${y}$ .
Find the vertex of the parabola by using the formula ${x = -\frac{b}{2a}}$ .
Plot the vertex and the ${y}$ -intercept on a coordinate plane.
Use the direction of the parabola to determine the shape of the graph.

Q: What are some common applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.
Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic equation solver that can be used to find the solutions to a quadratic equation.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not checking the solutions for extraneous solutions.
Not using the correct formula for the solutions.
Not simplifying the solutions.
Not checking the solutions for real and complex solutions.

Q: Can I use a quadratic equation to model a real-world problem?

A: Yes, you can use a quadratic equation to model a real-world problem. Quadratic equations can be used to model a wide range of phenomena, including the motion of objects, the behavior of economic systems, and the design of systems.