The Axis Of Symmetry For The Graph Of The Function F ( X ) = 3 X 2 + B X + 4 F(x)=3x^2+bx+4 F ( X ) = 3 X 2 + B X + 4 Is X = 3 2 X=\frac{3}{2} X = 2 3 ​ . What Is The Value Of B B B ?A. -18 B. -9 C. 9 D. 18

Understanding the Axis of Symmetry

The axis of symmetry is a concept in mathematics that plays a crucial role in the graph of quadratic functions. It is a vertical line that passes through the vertex of the parabola, which is the point where the function changes from decreasing to increasing or vice versa. The axis of symmetry is denoted by the equation $x = h$, where $h$ is the x-coordinate of the vertex.

The General Form of a Quadratic Function

A quadratic function can be written in the general form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola that opens upwards or downwards, depending on the sign of the coefficient $a$. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.

The Axis of Symmetry Formula

The axis of symmetry for a quadratic function in the form $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. This formula can be used to find the x-coordinate of the vertex of the parabola, which is the point where the axis of symmetry passes through.

Given Information

In this problem, we are given that the axis of symmetry for the graph of the function $f(x) = 3x^2 + bx + 4$ is $x = \frac{3}{2}$. We need to find the value of $b$.

Using the Axis of Symmetry Formula

We can use the axis of symmetry formula to find the value of $b$. Since the axis of symmetry is $x = \frac{3}{2}$, we can set up the equation $-\frac{b}{2a} = \frac{3}{2}$.

Solving for $b$

We can solve for $b$ by multiplying both sides of the equation by $-2a$. This gives us $b = -6a$. However, we need to find the value of $b$ in terms of the given function $f(x) = 3x^2 + bx + 4$.

Comparing Coefficients

We can compare the coefficients of the given function with the general form of a quadratic function. We have $a = 3$ and $c = 4$. We can substitute these values into the equation $b = -6a$ to find the value of $b$.

Finding the Value of $b$

Substituting $a = 3$ into the equation $b = -6a$, we get $b = -6(3) = -18$. Therefore, the value of $b$ is $-18$.

Conclusion

In this problem, we used the axis of symmetry formula to find the value of $b$ in the given quadratic function $f(x) = 3x^2 + bx + 4$. We compared the coefficients of the given function with the general form of a quadratic function and substituted the values into the equation $b = -6a$ to find the value of $b$. The final answer is $-18$.

Answer

The final answer is $\boxed{-18}$.

References

• [1] Axler, S. (2013). Linear Algebra Done Right. Springer.
• [2] Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.

Additional Resources

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-functions/x2f-quadratic-functions-article
• [2] Math Open Reference. (n.d.). Quadratic Functions. Retrieved from https://www.mathopenref.com/quadratic.html
  The Axis of Symmetry in Quadratic Functions: Q&A =====================================================

Q: What is the axis of symmetry in a quadratic function?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola, which is the point where the function changes from decreasing to increasing or vice versa.

Q: How is the axis of symmetry related to the vertex of a parabola?

A: The axis of symmetry passes through the vertex of the parabola. The vertex is the point where the function changes from decreasing to increasing or vice versa.

Q: What is the formula for the axis of symmetry?

A: The formula for the axis of symmetry is $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: How do I find the value of $b$ in a quadratic function using the axis of symmetry formula?

A: To find the value of $b$ using the axis of symmetry formula, you need to know the value of $a$ and the equation of the axis of symmetry. You can then substitute these values into the formula $x = -\frac{b}{2a}$ and solve for $b$.

Q: What is the significance of the axis of symmetry in quadratic functions?

A: The axis of symmetry is significant because it helps to identify the vertex of the parabola, which is the point where the function changes from decreasing to increasing or vice versa. This is useful in graphing and analyzing quadratic functions.

Q: Can the axis of symmetry be used to determine the direction of the parabola?

A: Yes, the axis of symmetry can be used to determine the direction of the parabola. If the axis of symmetry is to the left of the origin, the parabola opens to the left. If the axis of symmetry is to the right of the origin, the parabola opens to the right.

Q: How do I graph a quadratic function using the axis of symmetry?

A: To graph a quadratic function using the axis of symmetry, you need to know the equation of the axis of symmetry and the coordinates of the vertex. You can then use this information to draw the parabola.

Q: What are some real-world applications of the axis of symmetry in quadratic functions?

A: The axis of symmetry has many real-world applications, including:

• Physics: The axis of symmetry is used to describe the motion of objects under the influence of gravity.
• Engineering: The axis of symmetry is used to design and analyze structures such as bridges and buildings.
• Economics: The axis of symmetry is used to model and analyze economic systems.

Q: Can the axis of symmetry be used to solve quadratic equations?

A: Yes, the axis of symmetry can be used to solve quadratic equations. By using the axis of symmetry formula, you can find the x-coordinate of the vertex, which is the solution to the quadratic equation.

Q: What are some common mistakes to avoid when working with the axis of symmetry?

A: Some common mistakes to avoid when working with the axis of symmetry include:

• Not using the correct formula for the axis of symmetry.
• Not substituting the correct values into the formula.
• Not solving for the correct variable.

Q: How can I practice working with the axis of symmetry?

A: You can practice working with the axis of symmetry by:

• Graphing quadratic functions and identifying the axis of symmetry.
• Solving quadratic equations using the axis of symmetry formula.
• Analyzing and interpreting the results of your calculations.

Conclusion

The axis of symmetry is a fundamental concept in quadratic functions that has many real-world applications. By understanding the axis of symmetry, you can graph and analyze quadratic functions, solve quadratic equations, and make informed decisions in a variety of fields.