Find The { X$} − I N T E R C E P T S O F T H E P A R A B O L A G I V E N B Y T H E E Q U A T I O N . ( I F A N A N S W E R D O E S N O T E X I S T , E N T E R D N E . ) -intercepts Of The Parabola Given By The Equation. (If An Answer Does Not Exist, Enter DNE.) − In T Erce Pt So F T H E P A R Ab O L A G I V E Nb Y T H Ee Q U A T I O N . ( I F Anan S W Er D Oes N O T E X I S T , E N T ErD NE . ) ${ Y = 3x^2 - 15x }$1. { (x, Y) = (\square) $}$ (smaller { X$} − V A L U E ) 2. \[ -value)2. \[ − V A L U E ) 2. \[ (x, Y) = (\square)

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Introduction

In mathematics, the x-intercepts of a parabola are the points where the parabola intersects the x-axis. These points are also known as the roots or solutions of the quadratic equation that represents the parabola. In this article, we will focus on finding the x-intercepts of the parabola given by the equation y = 3x^2 - 15x.

Understanding the Equation

The given equation is a quadratic equation in the form of y = ax^2 + bx + c, where a = 3, b = -15, and c = 0. To find the x-intercepts, we need to set y = 0 and solve for x.

Setting y = 0

To find the x-intercepts, we set y = 0 and solve for x. This gives us the equation:

0 = 3x^2 - 15x

Factoring the Equation

We can factor out the common term x from the equation:

0 = x(3x - 15)

Solving for x

Now we have two possible solutions:

  1. x = 0
  2. 3x - 15 = 0

Solving the First Equation

The first equation is x = 0. This means that the x-intercept is at the point (0, 0).

Solving the Second Equation

The second equation is 3x - 15 = 0. We can solve for x by adding 15 to both sides of the equation:

3x = 15

Then, we can divide both sides of the equation by 3:

x = 5

Finding the x-Intercepts

Now that we have solved for x, we can find the x-intercepts of the parabola. The x-intercepts are the points where the parabola intersects the x-axis. In this case, the x-intercepts are at the points (0, 0) and (5, 0).

Conclusion

In conclusion, we have found the x-intercepts of the parabola given by the equation y = 3x^2 - 15x. The x-intercepts are at the points (0, 0) and (5, 0). We can use these points to graph the parabola and visualize its shape.

Discussion

  • What is the significance of the x-intercepts in the context of the parabola?
  • How do the x-intercepts relate to the roots or solutions of the quadratic equation?
  • Can you think of any real-world applications of finding the x-intercepts of a parabola?

Example Problems

  1. Find the x-intercepts of the parabola given by the equation y = 2x^2 + 4x - 3.
  2. Find the x-intercepts of the parabola given by the equation y = x^2 - 2x - 6.

Solutions

  1. To find the x-intercepts, we set y = 0 and solve for x:

0 = 2x^2 + 4x - 3

We can factor out the common term x from the equation:

0 = x(2x + 4 - 3)

Simplifying the equation, we get:

0 = x(2x + 1)

Now we have two possible solutions:

  1. x = 0
  2. 2x + 1 = 0

Solving the second equation, we get:

2x = -1

x = -1/2

The x-intercepts are at the points (0, 0) and (-1/2, 0).

  1. To find the x-intercepts, we set y = 0 and solve for x:

0 = x^2 - 2x - 6

We can factor the equation:

0 = (x - 3)(x + 2)

Now we have two possible solutions:

  1. x - 3 = 0
  2. x + 2 = 0

Solving the first equation, we get:

x = 3

Solving the second equation, we get:

x = -2

The x-intercepts are at the points (3, 0) and (-2, 0).

Final Answer

The final answer is:

  1. (0, 0)
  2. (5, 0)

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Q: What is the significance of the x-intercepts in the context of the parabola?

A: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. These points are also known as the roots or solutions of the quadratic equation that represents the parabola. The x-intercepts are significant because they provide information about the behavior of the parabola, such as its direction and shape.

Q: How do the x-intercepts relate to the roots or solutions of the quadratic equation?

A: The x-intercepts of a parabola are the same as the roots or solutions of the quadratic equation that represents the parabola. In other words, the x-intercepts are the values of x that make the quadratic equation equal to zero.

Q: Can you think of any real-world applications of finding the x-intercepts of a parabola?

A: Yes, there are many real-world applications of finding the x-intercepts of a parabola. For example, in physics, the x-intercepts of a parabola can be used to model the trajectory of a projectile, such as a thrown ball or a rocket. In engineering, the x-intercepts of a parabola can be used to design the shape of a curve or a surface.

Q: How do you find the x-intercepts of a parabola that is not in the form y = ax^2 + bx + c?

A: To find the x-intercepts of a parabola that is not in the form y = ax^2 + bx + c, you can use the following steps:

  1. Rewrite the equation in the form y = ax^2 + bx + c.
  2. Set y = 0 and solve for x.
  3. Use the quadratic formula to find the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the roots or solutions of a quadratic equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do you use the quadratic formula to find the x-intercepts of a parabola?

A: To use the quadratic formula to find the x-intercepts of a parabola, you can follow these steps:

  1. Identify the values of a, b, and c in the quadratic equation.
  2. Plug these values into the quadratic formula.
  3. Simplify the equation and solve for x.

Q: What are some common mistakes to avoid when finding the x-intercepts of a parabola?

A: Some common mistakes to avoid when finding the x-intercepts of a parabola include:

  • Not setting y = 0 before solving for x.
  • Not using the quadratic formula to find the x-intercepts.
  • Not simplifying the equation before solving for x.

Q: How do you graph a parabola using its x-intercepts?

A: To graph a parabola using its x-intercepts, you can follow these steps:

  1. Identify the x-intercepts of the parabola.
  2. Plot the x-intercepts on a coordinate plane.
  3. Draw a smooth curve through the x-intercepts to form the parabola.

Q: What are some real-world applications of graphing a parabola?

A: Some real-world applications of graphing a parabola include:

  • Modeling the trajectory of a projectile, such as a thrown ball or a rocket.
  • Designing the shape of a curve or a surface.
  • Analyzing the behavior of a system, such as a spring-mass system.

Q: How do you determine the direction and shape of a parabola?

A: To determine the direction and shape of a parabola, you can use the following steps:

  1. Identify the x-intercepts of the parabola.
  2. Determine the direction of the parabola by looking at the sign of the coefficient of x^2.
  3. Determine the shape of the parabola by looking at the value of the coefficient of x^2.

Q: What are some common types of parabolas?

A: Some common types of parabolas include:

  • Upward-facing parabolas: These parabolas open upward and have a positive coefficient of x^2.
  • Downward-facing parabolas: These parabolas open downward and have a negative coefficient of x^2.
  • Horizontal parabolas: These parabolas are horizontal and have a zero coefficient of x^2.

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

A: To find the vertex of a parabola, you can use the following steps:

  1. Identify the x-intercepts of the parabola.
  2. Determine the direction of the parabola by looking at the sign of the coefficient of x^2.
  3. Use the formula x = -b / 2a to find the x-coordinate of the vertex.
  4. Use the formula y = a(x - h)^2 + k to find the y-coordinate of the vertex.

Q: What are some real-world applications of finding the vertex of a parabola?

A: Some real-world applications of finding the vertex of a parabola include:

  • Modeling the trajectory of a projectile, such as a thrown ball or a rocket.
  • Designing the shape of a curve or a surface.
  • Analyzing the behavior of a system, such as a spring-mass system.

Q: How do you determine the axis of symmetry of a parabola?

A: To determine the axis of symmetry of a parabola, you can use the following steps:

  1. Identify the x-intercepts of the parabola.
  2. Determine the direction of the parabola by looking at the sign of the coefficient of x^2.
  3. Use the formula x = -b / 2a to find the x-coordinate of the axis of symmetry.

Q: What are some common mistakes to avoid when finding the axis of symmetry of a parabola?

A: Some common mistakes to avoid when finding the axis of symmetry of a parabola include:

  • Not identifying the x-intercepts of the parabola.
  • Not determining the direction of the parabola.
  • Not using the formula x = -b / 2a to find the x-coordinate of the axis of symmetry.

Q: How do you graph a parabola using its axis of symmetry?

A: To graph a parabola using its axis of symmetry, you can follow these steps:

  1. Identify the axis of symmetry of the parabola.
  2. Plot the axis of symmetry on a coordinate plane.
  3. Draw a smooth curve through the axis of symmetry to form the parabola.

Q: What are some real-world applications of graphing a parabola using its axis of symmetry?

A: Some real-world applications of graphing a parabola using its axis of symmetry include:

  • Modeling the trajectory of a projectile, such as a thrown ball or a rocket.
  • Designing the shape of a curve or a surface.
  • Analyzing the behavior of a system, such as a spring-mass system.

Q: How do you determine the equation of a parabola?

A: To determine the equation of a parabola, you can use the following steps:

  1. Identify the x-intercepts of the parabola.
  2. Determine the direction of the parabola by looking at the sign of the coefficient of x^2.
  3. Use the formula y = a(x - h)^2 + k to find the equation of the parabola.

Q: What are some common mistakes to avoid when determining the equation of a parabola?

A: Some common mistakes to avoid when determining the equation of a parabola include:

  • Not identifying the x-intercepts of the parabola.
  • Not determining the direction of the parabola.
  • Not using the formula y = a(x - h)^2 + k to find the equation of the parabola.

Q: How do you graph a parabola using its equation?

A: To graph a parabola using its equation, you can follow these steps:

  1. Identify the equation of the parabola.
  2. Plot the x-intercepts of the parabola on a coordinate plane.
  3. Draw a smooth curve through the x-intercepts to form the parabola.

Q: What are some real-world applications of graphing a parabola using its equation?

A: Some real-world applications of graphing a parabola using its equation include:

  • Modeling the trajectory of a projectile, such as a thrown ball or a rocket.
  • Designing the shape of a curve or a surface.
  • Analyzing the behavior of a system, such as a spring-mass system.

Q: How do you determine the vertex of a parabola using its equation?

A: To determine the vertex of a parabola using its equation, you can use the following steps:

  1. Identify the equation of the parabola.
  2. Use the formula x = -b / 2a to find the x-coordinate of the vertex.
  3. Use the formula y =