Determining the Direction of the Graph

Oct 13, 2025 by windailys.com 80 views

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 focus on the graph of the quadratic function f(x) = 2x^2 - x + 4 and determine whether its graph opens up or down, find its vertex, and axis of symmetry.

Determining the Direction of the Graph

To determine whether the graph of a quadratic function opens up or down, we need to look at the coefficient of the x^2 term, which is a. If a is positive, the graph opens up, and if a is negative, the graph opens down.

In the given function f(x) = 2x^2 - x + 4, the coefficient of the x^2 term is 2, which is positive. Therefore, the graph of the function opens up.

Finding the Vertex

The vertex of a quadratic function is the maximum or minimum point of the graph. To find the vertex, we need to use the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively.

In the given function f(x) = 2x^2 - x + 4, the coefficient of the x^2 term is 2, and the coefficient of the x term is -1. Plugging these values into the formula, we get x = -(-1) / (2*2) = 1/4.

To find the y-coordinate of the vertex, we need to plug the x-coordinate into the function. Substituting x = 1/4 into the function f(x) = 2x^2 - x + 4, we get f(1/4) = 2(1/4)^2 - (1/4) + 4 = 2(1/16) - 1/4 + 4 = 1/8 - 1/4 + 4 = 1/8 - 2/8 + 32/8 = 31/8.

Therefore, the coordinates of the vertex are (1/4, 31/8).

Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the graph. To find the axis of symmetry, we need to use the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively.

In the given function f(x) = 2x^2 - x + 4, the coefficient of the x^2 term is 2, and the coefficient of the x term is -1. Plugging these values into the formula, we get x = -(-1) / (2*2) = 1/4.

Therefore, the equation of the axis of symmetry is x = 1/4.

Conclusion

In conclusion, the graph of the quadratic function f(x) = 2x^2 - x + 4 opens up, and its vertex is located at (1/4, 31/8). The axis of symmetry of the graph is the vertical line x = 1/4.

Key Takeaways

The graph of a quadratic function opens up if the coefficient of the x^2 term is positive and opens down if the coefficient is negative.
The vertex of a quadratic function is the maximum or minimum point of the graph.
The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the graph.

Practice Problems

Determine whether the graph of the quadratic function f(x) = 3x^2 + 2x - 1 opens up or down.
Find the vertex and axis of symmetry of the quadratic function f(x) = 2x^2 - 3x + 1.
Determine whether the graph of the quadratic function f(x) = x^2 - 4x + 3 opens up or down.

Solutions

The graph of the quadratic function f(x) = 3x^2 + 2x - 1 opens up because the coefficient of the x^2 term is positive.
The vertex of the quadratic function f(x) = 2x^2 - 3x + 1 is located at (3/4, -5/8), and the axis of symmetry is the vertical line x = 3/4.
The graph of the quadratic function f(x) = x^2 - 4x + 3 opens down because the coefficient of the x^2 term is positive.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Graphing Quadratic Functions" by Purplemath
[3] "Quadratic Equations and Functions" by Khan Academy
Quadratic Function Q&A ==========================

Frequently Asked Questions

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: How do I determine whether the graph of a quadratic function opens up or down?

A: To determine whether the graph of a quadratic function opens up or down, you need to look at the coefficient of the x^2 term, which is a. If a is positive, the graph opens up, and if a is negative, the graph opens down.

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

A: To find the vertex of a quadratic function, you need to use the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. Then, you need to plug the x-coordinate into the function to find the y-coordinate.

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

A: To find the axis of symmetry of a quadratic function, you need to use the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. This will give you the x-coordinate of the axis of symmetry.

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

A: The axis of symmetry is a vertical line that passes through the vertex of the graph. It is a line of symmetry, meaning that the graph is symmetric about this line.

Q: Can a quadratic function have more than one axis of symmetry?

A: No, a quadratic function can only have one axis of symmetry.

Q: Can a quadratic function have a vertex that is not on the axis of symmetry?

A: No, the vertex of a quadratic function must be on the axis of symmetry.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the vertex and axis of symmetry, and then plot points on either side of the axis of symmetry. You can use a graphing calculator or software to graph the function.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient, which means the graph will open down.

Q: Can a quadratic function have a zero leading coefficient?

A: No, a quadratic function cannot have a zero leading coefficient, as this would make it a linear function.

Q: Can a quadratic function have a negative constant term?

A: Yes, a quadratic function can have a negative constant term.

Q: Can a quadratic function have a zero constant term?

A: Yes, a quadratic function can have a zero constant term.

Q: Can a quadratic function have a negative coefficient of the x term?

A: Yes, a quadratic function can have a negative coefficient of the x term.

Q: Can a quadratic function have a zero coefficient of the x term?

A: Yes, a quadratic function can have a zero coefficient of the x term.

Additional Resources

[1] "Quadratic Functions" by Math Open Reference
[2] "Graphing Quadratic Functions" by Purplemath
[3] "Quadratic Equations and Functions" by Khan Academy

Practice Problems

Determine whether the graph of the quadratic function f(x) = 3x^2 + 2x - 1 opens up or down.
Find the vertex and axis of symmetry of the quadratic function f(x) = 2x^2 - 3x + 1.
Graph the quadratic function f(x) = x^2 - 4x + 3.

Solutions

The graph of the quadratic function f(x) = 3x^2 + 2x - 1 opens up because the coefficient of the x^2 term is positive.
The vertex of the quadratic function f(x) = 2x^2 - 3x + 1 is located at (3/4, -5/8), and the axis of symmetry is the vertical line x = 3/4.
The graph of the quadratic function f(x) = x^2 - 4x + 3 opens down because the coefficient of the x^2 term is positive.