Find The Equation Of The Axis Of Symmetry For The Quadratic Function:$\[ Y = -5x^2 - 25x - 48 \\]Answer: $\[ X = \square \\]

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Introduction

In mathematics, the axis of symmetry is a line that passes through the vertex of a parabola and divides it into two equal parts. It is an essential concept in algebra, particularly when dealing with quadratic functions. In this article, we will focus on finding the equation of the axis of symmetry for a given quadratic function.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the general form:

y=ax2+bx+c{ y = ax^2 + bx + c }

where a, b, and c are constants, and x is the variable. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. The vertex is the point on the parabola where it changes direction, and it is the lowest or highest point on the graph. The equation of the axis of symmetry is given by:

x=βˆ’b2a{ x = -\frac{b}{2a} }

This equation is derived from the fact that the axis of symmetry passes through the vertex, which is the point where the parabola changes direction.

Finding the Equation of the Axis of Symmetry

To find the equation of the axis of symmetry, we need to identify the values of a and b in the quadratic function. In the given function:

y=βˆ’5x2βˆ’25xβˆ’48{ y = -5x^2 - 25x - 48 }

we can see that a = -5 and b = -25.

Step 1: Identify the Values of a and b

The values of a and b are -5 and -25, respectively.

Step 2: Plug the Values into the Equation

Now that we have the values of a and b, we can plug them into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

Substituting the values, we get:

x=βˆ’βˆ’252(βˆ’5){ x = -\frac{-25}{2(-5)} }

Simplify the Equation

Simplifying the equation, we get:

x=βˆ’βˆ’25βˆ’10{ x = -\frac{-25}{-10} }

x=25βˆ’10{ x = \frac{25}{-10} }

x=βˆ’52{ x = -\frac{5}{2} }

Conclusion

In this article, we have discussed the concept of the axis of symmetry and how to find its equation for a given quadratic function. We have used the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

to find the equation of the axis of symmetry for the given function:

y=βˆ’5x2βˆ’25xβˆ’48{ y = -5x^2 - 25x - 48 }

The final answer is:

x=βˆ’52{ x = -\frac{5}{2} }

This equation represents the axis of symmetry for the given quadratic function.

Example Problems

Problem 1

Find the equation of the axis of symmetry for the quadratic function:

y=2x2+6x+5{ y = 2x^2 + 6x + 5 }

Solution

To find the equation of the axis of symmetry, we need to identify the values of a and b in the quadratic function. In this function, a = 2 and b = 6.

Plugging the values into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

we get:

x=βˆ’62(2){ x = -\frac{6}{2(2)} }

Simplifying the equation, we get:

x=βˆ’64{ x = -\frac{6}{4} }

x=βˆ’32{ x = -\frac{3}{2} }

Problem 2

Find the equation of the axis of symmetry for the quadratic function:

y=βˆ’3x2βˆ’9x+2{ y = -3x^2 - 9x + 2 }

Solution

To find the equation of the axis of symmetry, we need to identify the values of a and b in the quadratic function. In this function, a = -3 and b = -9.

Plugging the values into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

we get:

x=βˆ’βˆ’92(βˆ’3){ x = -\frac{-9}{2(-3)} }

Simplifying the equation, we get:

x=βˆ’βˆ’9βˆ’6{ x = -\frac{-9}{-6} }

x=9βˆ’6{ x = \frac{9}{-6} }

x=βˆ’32{ x = -\frac{3}{2} }

Final Answer

The final answer is:

x=βˆ’52{ x = -\frac{5}{2} }

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Introduction

In our previous article, we discussed how to find the equation of the axis of symmetry for a given quadratic function. In this article, we will answer some frequently asked questions about the axis of symmetry and provide additional examples to help you understand the concept better.

Q&A

Q1: What is the axis of symmetry?

A1: The axis of symmetry is a vertical line that passes through the vertex of a parabola and divides it into two equal parts.

Q2: How do I find the equation of the axis of symmetry?

A2: To find the equation of the axis of symmetry, you need to identify the values of a and b in the quadratic function and plug them into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

Q3: What if the quadratic function has a negative value for a?

A3: If the quadratic function has a negative value for a, you will get a negative value for the equation of the axis of symmetry. For example, if a = -5 and b = -25, the equation of the axis of symmetry will be:

x=βˆ’βˆ’252(βˆ’5){ x = -\frac{-25}{2(-5)} }

x=βˆ’βˆ’25βˆ’10{ x = -\frac{-25}{-10} }

x=25βˆ’10{ x = \frac{25}{-10} }

x=βˆ’52{ x = -\frac{5}{2} }

Q4: Can the axis of symmetry be a horizontal line?

A4: No, the axis of symmetry cannot be a horizontal line. It is always a vertical line that passes through the vertex of the parabola.

Q5: How do I know if the axis of symmetry is a positive or negative value?

A5: To determine if the axis of symmetry is a positive or negative value, you need to look at the value of a in the quadratic function. If a is positive, the axis of symmetry will be a positive value. If a is negative, the axis of symmetry will be a negative value.

Q6: Can I use the axis of symmetry to find the vertex of the parabola?

A6: Yes, you can use the axis of symmetry to find the vertex of the parabola. The vertex is the point on the parabola where it changes direction, and it is the lowest or highest point on the graph. The axis of symmetry passes through the vertex, so you can use it to find the coordinates of the vertex.

Example Problems

Problem 1

Find the equation of the axis of symmetry for the quadratic function:

y=3x2+12x+15{ y = 3x^2 + 12x + 15 }

Solution

To find the equation of the axis of symmetry, we need to identify the values of a and b in the quadratic function. In this function, a = 3 and b = 12.

Plugging the values into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

we get:

x=βˆ’122(3){ x = -\frac{12}{2(3)} }

Simplifying the equation, we get:

x=βˆ’126{ x = -\frac{12}{6} }

x=βˆ’2{ x = -2 }

Problem 2

Find the equation of the axis of symmetry for the quadratic function:

y=βˆ’2x2βˆ’8x+5{ y = -2x^2 - 8x + 5 }

Solution

To find the equation of the axis of symmetry, we need to identify the values of a and b in the quadratic function. In this function, a = -2 and b = -8.

Plugging the values into the equation:

x=βˆ’b2a{ x = -\frac{b}{2a} }

we get:

x=βˆ’βˆ’82(βˆ’2){ x = -\frac{-8}{2(-2)} }

Simplifying the equation, we get:

x=βˆ’βˆ’8βˆ’4{ x = -\frac{-8}{-4} }

x=8βˆ’4{ x = \frac{8}{-4} }

x=βˆ’2{ x = -2 }

Final Answer

The final answer is:

x=βˆ’2{ x = -2 }

This equation represents the axis of symmetry for the given quadratic function.

Conclusion

In this article, we have answered some frequently asked questions about the axis of symmetry and provided additional examples to help you understand the concept better. We have also discussed how to find the equation of the axis of symmetry for a given quadratic function.