Plot Five Points On The Graph Of The Function G ( X ) = − 4 X 2 + 4 G(x)=-4x^2+4 G ( X ) = − 4 X 2 + 4 :1. One Point With $x=0$2. Two Points With Negative X X X -values3. Two Points With Positive X X X -valuesThen Click On The Graph-a-function Button.

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 quadratic function $g(x) = -4x^2 + 4$ and plot five points on its graph.

Understanding the Function

The given function is $g(x) = -4x^2 + 4$. This function is a quadratic function because the highest power of the variable $x$ is two. The coefficient of $x^2$ is $-4$, which is negative, indicating that the parabola will open downwards. The constant term is $4$, which is positive and will shift the parabola upwards.

Plotting Points

To plot points on the graph of the function $g(x) = -4x^2 + 4$, we need to find the values of $x$ and $y$ for each point. We will start by finding the point with $x = 0$.

Point 1: $x = 0$

To find the point with $x = 0$, we substitute $x = 0$ into the function $g(x) = -4x^2 + 4$.

import numpy as np
def g(x):
return -4*x**2 + 4

x = 0
y = g(x)
print(f"Point 1: ({x}, {y})")

When we run this code, we get the point $(0, 4)$.

Points 2 and 3: Negative $x$-values

To find the points with negative $x$-values, we can substitute $x = -1$ and $x = -2$ into the function $g(x) = -4x^2 + 4$.

# Find the points with negative x-values
x = -1
y = g(x)
print(f"Point 2: ({x}, {y})")
x = -2
y = g(x)
print(f"Point 3: ({x}, {y})")

When we run this code, we get the points $(-1, 6)$ and $(-2, 12)$.

Points 4 and 5: Positive $x$-values

To find the points with positive $x$-values, we can substitute $x = 1$ and $x = 2$ into the function $g(x) = -4x^2 + 4$.

# Find the points with positive x-values
x = 1
y = g(x)
print(f"Point 4: ({x}, {y})")
x = 2
y = g(x)
print(f"Point 5: ({x}, {y})")

When we run this code, we get the points $(1, 2)$ and $(2, 0)$.

Conclusion

In this article, we plotted five points on the graph of the quadratic function $g(x) = -4x^2 + 4$. We found the points with $x = 0$, negative $x$-values, and positive $x$-values. The points we found are $(0, 4)$, $(-1, 6)$, $(-2, 12)$, $(1, 2)$, and $(2, 0)$. These points can be used to visualize the graph of the function and understand its behavior.

Graph of the Function

Here is the graph of the function $g(x) = -4x^2 + 4$ with the five points we plotted:

import matplotlib.pyplot as plt
import numpy as np

def g(x):
return -4*x**2 + 4

x = np.linspace(-3, 3, 400)

y = g(x)

plt.plot(x, y)
plt.scatter([0, -1, -2, 1, 2], [4, 6, 12, 2, 0])
plt.title("Graph of the Function g(x) = -4x^2 + 4")
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

$$

Introduction

In our previous article, we explored the quadratic function $g(x) = -4x^2 + 4$ and plotted five points on its graph. In this article, we will answer some frequently asked questions about quadratic function graphs.

Q: What is a quadratic function?

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 graph of a quadratic function?

The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the sign of the coefficient of $x^2$. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

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

The vertex of a quadratic function is the point on the parabola that is lowest or highest. To find the vertex, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of $x^2$ and $x$, respectively.

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

The x-intercepts of a quadratic function are the points on the parabola where the graph crosses the x-axis. To find the x-intercepts, you can set the function equal to zero and solve for $x$.

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

The y-intercept of a quadratic function is the point on the parabola where the graph crosses the y-axis. To find the y-intercept, you can substitute $x = 0$ into the function.

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

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The axis of symmetry is given by the equation $x = -\frac{b}{2a}$.

Q: How do I graph a quadratic function?

To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot points on the graph.

Q: What are some real-world applications of quadratic functions?

Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Solving optimization problems
• Modeling population growth or decline
• Finding the area or perimeter of a shape

Conclusion

In this article, we answered some frequently asked questions about quadratic function graphs. We discussed the definition of a quadratic function, the graph of a quadratic function, and how to find the vertex, x-intercepts, y-intercept, and axis of symmetry of a quadratic function. We also discussed some real-world applications of quadratic functions.

Additional Resources

For more information on quadratic functions, you can check out the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Wolfram Alpha: Quadratic Functions

Practice Problems

Here are some practice problems to help you review quadratic functions:

1. Find the vertex of the quadratic function $f(x) = x^2 + 3x + 2$.
2. Find the x-intercepts of the quadratic function $f(x) = x^2 - 4x + 3$.
3. Find the y-intercept of the quadratic function $f(x) = 2x^2 - 5x + 1$.
4. Find the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$.
5. Graph the quadratic function $f(x) = x^2 - 2x + 1$.

I hope this article has been helpful in answering your questions about quadratic function graphs. If you have any further questions, please don't hesitate to ask.