Understanding the Graph of the Quadratic Function

Oct 13, 2025 by miodeya.com 96 views

Analyzing the Quadratic Function $f(x) = 3x^2 + 7x + 2$

In this article, we will delve into the world of quadratic functions and explore the properties of the given function $f(x) = 3x^2 + 7x + 2$ . We will calculate the value of the discriminant, determine the number of $x$ -intercepts, and find the number of zeros for this function.

1. What is the value of the discriminant?

The discriminant of a quadratic function $f(x) = ax^2 + bx + c$ is given by the formula $b^2 - 4ac$ . In our case, $a = 3$ , $b = 7$ , and $c = 2$ . To find the value of the discriminant, we will substitute these values into the formula.

import math

# Define the coefficients of the quadratic function
a = 3
b = 7
c = 2

# Calculate the discriminant
discriminant = b**2 - 4*a*c

print("The value of the discriminant is:", discriminant)

When we run this code, we get the value of the discriminant as $49 - 24 = 25$ . Therefore, the value of the discriminant is $25$ .

2. How many $x$ -intercepts does this function have?

The number of $x$ -intercepts of a quadratic function is determined by the value of the discriminant. If the discriminant is positive, the function has two $x$ -intercepts. If the discriminant is zero, the function has one $x$ -intercept. If the discriminant is negative, the function has no $x$ -intercepts.

Since the discriminant is $25$ , which is positive, the function $f(x) = 3x^2 + 7x + 2$ has two $x$ -intercepts.

3. What is the number of zeros for this function?

The number of zeros of a quadratic function is equal to the number of $x$ -intercepts. Since we have already determined that the function has two $x$ -intercepts, the number of zeros for this function is also two.

Understanding the Graph of the Quadratic Function

The graph of a quadratic function is a parabola that opens upward or downward. The vertex of the parabola is the minimum or maximum point of the function. The $x$ -intercepts of the function are the points where the parabola intersects the $x$ -axis.

To graph the function $f(x) = 3x^2 + 7x + 2$ , we can use the following steps:

Find the vertex of the parabola by using the formula $x = -\frac{b}{2a}$ .
Plot the vertex on the graph.
Determine the direction of the parabola by examining the sign of the coefficient of the $x^2$ term.
Plot the $x$ -intercepts on the graph.

import matplotlib.pyplot as plt
import numpy as np

# Define the coefficients of the quadratic function
a = 3
b = 7
c = 2

# Create an array of x values
x = np.linspace(-10, 10, 400)

# Calculate the corresponding y values
y = a*x**2 + b*x + c

# Plot the graph
plt.plot(x, y)
plt.title("Graph of the Quadratic Function")
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

When we run this code, we get the graph of the quadratic function.

Conclusion

In this article, we analyzed the quadratic function $f(x) = 3x^2 + 7x + 2$ and determined the value of the discriminant, the number of $x$ -intercepts, and the number of zeros for this function. We also graphed the function using Python code. The graph of the function is a parabola that opens upward, and it has two $x$ -intercepts.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Graphing Quadratic Functions" by Purplemath
[3] "Quadratic Formula" by Wolfram MathWorld
Quadratic Function Q&A

In this article, we will answer some frequently asked questions about quadratic functions. We will cover topics such as the discriminant, $x$ -intercepts, zeros, and graphing quadratic functions.

Q: What is the discriminant of a quadratic function?

A: The discriminant of a quadratic function $f(x) = ax^2 + bx + c$ is given by the formula $b^2 - 4ac$ . It is used to determine the number of $x$ -intercepts of the function.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to substitute the values of $a$ , $b$ , and $c$ into the formula $b^2 - 4ac$ . For example, if $a = 3$ , $b = 7$ , and $c = 2$ , the discriminant would be $7^2 - 4(3)(2) = 49 - 24 = 25$ .

Q: What does the discriminant tell me about the function?

A: The discriminant tells you the number of $x$ -intercepts of the function. If the discriminant is positive, the function has two $x$ -intercepts. If the discriminant is zero, the function has one $x$ -intercept. If the discriminant is negative, the function has no $x$ -intercepts.

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 solve the equation $f(x) = 0$ . This can be done using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: What is the difference between $x$ -intercepts and zeros?

A: $x$ -intercepts and zeros are the same thing. They are the points where the graph of the function intersects the $x$ -axis.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to follow these steps:

Find the vertex of the parabola by using the formula $x = -\frac{b}{2a}$ .
Plot the vertex on the graph.
Determine the direction of the parabola by examining the sign of the coefficient of the $x^2$ term.
Plot the $x$ -intercepts on the graph.

Q: Can I use a calculator to graph a quadratic function?

A: Yes, you can use a calculator to graph a quadratic function. Most graphing calculators have a built-in function for graphing quadratic functions.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the minimum or maximum point of the function. It is the point where the parabola changes direction.

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 $x = -\frac{b}{2a}$ .

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is the line that divides the parabola into two equal parts.

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 $x = -\frac{b}{2a}$ .

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We covered topics such as the discriminant, $x$ -intercepts, zeros, and graphing quadratic functions. We hope that this article has been helpful in understanding quadratic functions.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Graphing Quadratic Functions" by Purplemath
[3] "Quadratic Formula" by Wolfram MathWorld