Select The Correct Answer.Consider The Given Function:$ F(x) = X^2 - 14x - 72 }$What Are The Zeros And Axis Of Symmetry For The Graph Of The Function?A. Zeros { X = -4$ $ And { X = 18$} ; A X I S O F S Y M M E T R Y : \[ ; Axis Of Symmetry: \[ ; A X I So F Sy Mm E T Ry : \[ X =

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Introduction

In mathematics, quadratic equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable. In this article, we will focus on solving quadratic equations of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c and finding the zeros and axis of symmetry for the graph of the function.

The Given Function

The given function is f(x)=x2−14x−72f(x) = x^2 - 14x - 72. To find the zeros and axis of symmetry, we need to factorize the quadratic expression.

Factoring the Quadratic Expression

To factorize the quadratic expression, we need to find two numbers whose product is −72-72 and whose sum is −14-14. These numbers are −18-18 and 44, because −18×4=−72-18 \times 4 = -72 and −18+4=−14-18 + 4 = -14. Therefore, we can write the quadratic expression as:

f(x)=(x−18)(x+4)f(x) = (x - 18)(x + 4)

Finding the Zeros

The zeros of a quadratic function are the values of xx that make the function equal to zero. In other words, we need to find the values of xx that satisfy the equation f(x)=0f(x) = 0. To do this, we can set each factor equal to zero and solve for xx.

x−18=0⇒x=18x - 18 = 0 \Rightarrow x = 18

x+4=0⇒x=−4x + 4 = 0 \Rightarrow x = -4

Therefore, the zeros of the function are x=−4x = -4 and x=18x = 18.

Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. To find the axis of symmetry, we need to find the value of xx that makes the function equal to its average value. This value is given by the formula:

x=−b2ax = -\frac{b}{2a}

In this case, a=1a = 1 and b=−14b = -14, so we have:

x=−−142(1)=142=7x = -\frac{-14}{2(1)} = \frac{14}{2} = 7

Therefore, the axis of symmetry is the vertical line x=7x = 7.

Conclusion

In this article, we have solved a quadratic equation of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c and found the zeros and axis of symmetry for the graph of the function. We have shown that the zeros of the function are x=−4x = -4 and x=18x = 18, and the axis of symmetry is the vertical line x=7x = 7. These results are important in various fields, including algebra, geometry, and calculus.

Key Takeaways

  • The zeros of a quadratic function are the values of xx that make the function equal to zero.
  • The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola.
  • To find the zeros and axis of symmetry, we need to factorize the quadratic expression and use the formulas x=−b2ax = -\frac{b}{2a} and x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Practice Problems

  1. Find the zeros and axis of symmetry for the graph of the function f(x)=x2+6x+8f(x) = x^2 + 6x + 8.
  2. Find the zeros and axis of symmetry for the graph of the function f(x)=x2−2x−15f(x) = x^2 - 2x - 15.
  3. Find the zeros and axis of symmetry for the graph of the function f(x)=x2+5x+6f(x) = x^2 + 5x + 6.

Solutions

  1. The zeros of the function are x=−4x = -4 and x=−2x = -2, and the axis of symmetry is the vertical line x=−3x = -3.
  2. The zeros of the function are x=−5x = -5 and x=3x = 3, and the axis of symmetry is the vertical line x=−1x = -1.
  3. The zeros of the function are x=−3x = -3 and x=−2x = -2, and the axis of symmetry is the vertical line x=−52x = -\frac{5}{2}.

References

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for success in various fields, including algebra, geometry, and calculus. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their applications.

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. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following methods:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for xx.
  • Quadratic Formula: If the quadratic expression cannot be factored, you can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • Graphing: You can also graph the quadratic function and find the zeros by finding the points where the graph intersects the x-axis.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. To find the axis of symmetry, you can use the formula: x=−b2ax = -\frac{b}{2a}.

Q: What are the zeros of a quadratic function?

A: The zeros of a quadratic function are the values of xx that make the function equal to zero. In other words, you need to find the values of xx that satisfy the equation f(x)=0f(x) = 0. To do this, you can set each factor equal to zero and solve for xx.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula: x=−b2ax = -\frac{b}{2a}. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can plug the x-coordinate into the quadratic function.

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. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is used to solve quadratic equations, while linear equations can be solved using other methods, such as factoring or graphing.

Q: What are some real-world 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 bridges, buildings, and other structures.
  • Economics: Quadratic equations are used to model the behavior of economic systems.
  • Computer Science: Quadratic equations are used in algorithms and data structures.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand quadratic equations and their applications. We have covered topics such as solving quadratic equations, finding the axis of symmetry, and real-world applications of quadratic equations. We hope this guide has been helpful in your understanding of quadratic equations.

Practice Problems

  1. Solve the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0.
  2. Find the axis of symmetry for the quadratic function f(x)=x2−2x−15f(x) = x^2 - 2x - 15.
  3. Find the zeros of the quadratic function f(x)=x2+3x+2f(x) = x^2 + 3x + 2.

Solutions

  1. The solutions to the quadratic equation are x=−2x = -2 and x=−3x = -3.
  2. The axis of symmetry is the vertical line x=−22=−1x = -\frac{2}{2} = -1.
  3. The zeros of the quadratic function are x=−1x = -1 and x=−2x = -2.

References