Given: $f(x) = 2x^2 - 8$1. Calculate The $x$-intercepts Of $f$.2. Calculate The $y$-intercept Of $f$.3. Write Down The Coordinates Of The Turning Point Of $f$.4. Sketch The Graph Of $f$,

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Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will explore the graph of a quadratic function given by f(x) = 2x^2 - 8.

Calculating the x-Intercepts

The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. To find the x-intercepts, we need to set the function equal to zero and solve for x.

Given: f(x) = 2x^2 - 8

Set f(x) = 0:

2x^2 - 8 = 0

Add 8 to both sides:

2x^2 = 8

Divide both sides by 2:

x^2 = 4

Take the square root of both sides:

x = ±√4

x = Β±2

Therefore, the x-intercepts of f are (2, 0) and (-2, 0).

Calculating the y-Intercept

The y-intercept of a quadratic function is the point where the graph of the function crosses the y-axis. To find the y-intercept, we need to evaluate the function at x = 0.

Given: f(x) = 2x^2 - 8

Evaluate f(0):

f(0) = 2(0)^2 - 8

f(0) = -8

Therefore, the y-intercept of f is (0, -8).

Finding the Turning Point

The turning point of a quadratic function is the point where the graph of the function changes direction. To find the turning point, we need to find the vertex of the parabola.

The x-coordinate of the vertex is given by:

x = -b / 2a

In this case, a = 2 and b = 0:

x = -0 / 2(2)

x = 0

To find the y-coordinate of the vertex, we need to evaluate the function at x = 0:

f(0) = 2(0)^2 - 8

f(0) = -8

Therefore, the coordinates of the turning point are (0, -8).

Sketching the Graph

To sketch the graph of f, we need to plot the x-intercepts, y-intercept, and turning point.

  • Plot the x-intercepts (2, 0) and (-2, 0)
  • Plot the y-intercept (0, -8)
  • Plot the turning point (0, -8)
  • Draw a smooth curve through the points

The graph of f is a parabola that opens upwards, with the x-intercepts at (2, 0) and (-2, 0), the y-intercept at (0, -8), and the turning point at (0, -8).

Conclusion

In this article, we have explored the graph of a quadratic function given by f(x) = 2x^2 - 8. We have calculated the x-intercepts, y-intercept, and turning point of the function, and sketched the graph of the function. The graph of f is a parabola that opens upwards, with the x-intercepts at (2, 0) and (-2, 0), the y-intercept at (0, -8), and the turning point at (0, -8).

Key Takeaways

  • The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis.
  • The y-intercept of a quadratic function is the point where the graph of the function crosses the y-axis.
  • The turning point of a quadratic function is the point where the graph of the function changes direction.
  • The graph of a quadratic function is a parabola that opens upwards or downwards.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

  • Khan Academy: Quadratic Functions
  • Math Is Fun: Quadratic Functions
  • Wolfram MathWorld: Quadratic Function

References

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

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

A: A linear function is a polynomial function of degree one, which means the highest power of the variable is one. The general form of a linear function is f(x) = ax + b, where a and b are constants. A quadratic function, on the other hand, is a polynomial function of degree two.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x. This will give you the points where the graph of the function crosses the x-axis.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you need to evaluate the function at x = 0. This will give you the point where the graph of the function crosses the y-axis.

Q: What is the turning point of a quadratic function?

A: The turning point of a quadratic function is the point where the graph of the function changes direction. It is also known as the vertex of the parabola.

Q: How do I find the turning point of a quadratic function?

A: To find the turning point of a quadratic function, you need to find the vertex of the parabola. The x-coordinate of the vertex is given by x = -b / 2a, and the y-coordinate is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex.

Q: What is the graph of a quadratic function like?

A: The graph of a quadratic function is a parabola that opens upwards or downwards. The parabola can be symmetrical or asymmetrical, depending on the value of a.

Q: Can a quadratic function have more than one turning point?

A: No, a quadratic function can only have one turning point. The turning point is the vertex of the parabola, and it is the point where the graph of the function changes direction.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the graph of the function does not cross the x-axis.

Q: Can a quadratic function have no y-intercept?

A: No, a quadratic function cannot have no y-intercept. The y-intercept is the point where the graph of the function crosses the y-axis, and it is always present.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This will give you the solutions to the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.

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

A: Yes, you can use the quadratic formula to solve any quadratic equation. However, you need to make sure that the equation is in the form ax^2 + bx + c = 0.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many common applications in mathematics, science, and engineering. Some examples include:

  • Modeling the trajectory of a projectile
  • Finding the maximum or minimum value of a function
  • Solving systems of equations
  • Finding the area or perimeter of a shape

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic functions. We hope that this article has been helpful in understanding the concept of quadratic functions and how to work with them.