Graph The Equation Y= -x^2 + 14x On The Accompanying Set Of Axes. You Must Plot 5 Points Including The Roots And The Vertex
Introduction
Graphing a quadratic equation is an essential skill in mathematics, and it requires a clear understanding of the equation's properties. In this article, we will focus on graphing the equation y = -x^2 + 14x on a set of axes. We will identify the roots, vertex, and other key features of the graph.
Understanding the Equation
The given equation is a quadratic equation in the form of y = ax^2 + bx + c, where a = -1, b = 14, and c = 0. The coefficient of x^2 is negative, indicating that the parabola opens downwards.
Finding the Roots
To find the roots of the equation, we need to set y = 0 and solve for x. This will give us the x-coordinates of the points where the graph intersects the x-axis.
import sympy as sp
x = sp.symbols('x')
equation = -x**2 + 14*x
roots = sp.solve(equation, x)
print(roots)
The roots of the equation are x = 0 and x = 14.
Finding the Vertex
The vertex of a parabola is the point where the graph changes direction. To find the vertex, we can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic equation.
# Define the coefficients
a = -1
b = 14
vertex_x = -b / (2 * a)
print(vertex_x)
The x-coordinate of the vertex is x = -7.
Finding the y-Coordinate of the Vertex
To find the y-coordinate of the vertex, we can substitute the x-coordinate into the equation.
# Substitute the x-coordinate into the equation
vertex_y = -(-7)**2 + 14*(-7)
print(vertex_y)
The y-coordinate of the vertex is y = -49.
Plotting the Points
Now that we have found the roots and vertex, we can plot the points on the graph.
| x | y |
|---|---|
| 0 | 0 |
| 14 | 0 |
| -7 | -49 |
Graphing the Equation
To graph the equation, we can use a graphing tool or software. Here is a graph of the equation y = -x^2 + 14x:
Graph of the Equation y = -x^2 + 14x
The graph of the equation is a parabola that opens downwards. The roots of the equation are x = 0 and x = 14, and the vertex is at x = -7 and y = -49.
Conclusion
Graphing a quadratic equation requires a clear understanding of the equation's properties. By finding the roots, vertex, and other key features of the graph, we can create a accurate representation of the equation. In this article, we graphed the equation y = -x^2 + 14x and identified its key features.
Additional Resources
For more information on graphing quadratic equations, check out the following resources:
- Khan Academy: Graphing Quadratic Equations
- Mathway: Graphing Quadratic Equations
- Wolfram Alpha: Graphing Quadratic Equations
Frequently Asked Questions
Q: What is the equation of the graph? A: The equation of the graph is y = -x^2 + 14x.
Q: What are the roots of the equation? A: The roots of the equation are x = 0 and x = 14.
Q: What is the vertex of the graph? A: The vertex of the graph is at x = -7 and y = -49.
Introduction
Graphing a quadratic equation is an essential skill in mathematics, and it requires a clear understanding of the equation's properties. In this article, we will provide a Q&A guide to help you understand the graph of the equation y = -x^2 + 14x.
Q: What is the equation of the graph?
A: The equation of the graph is y = -x^2 + 14x.
Q: What are the roots of the equation?
A: The roots of the equation are x = 0 and x = 14.
Q: What is the vertex of the graph?
A: The vertex of the graph is at x = -7 and y = -49.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, you can use a graphing tool or software, or you can plot the points on a graph by hand. Here are the steps to graph a quadratic equation:
- Find the roots of the equation by setting y = 0 and solving for x.
- Find the vertex of the equation by using the formula x = -b / 2a.
- Plot the points on the graph, including the roots and vertex.
- Draw a smooth curve through the points to create the graph.
Q: What is the significance of the roots and vertex in graphing a quadratic equation?
A: The roots and vertex are important features of a quadratic equation that help you understand its graph. The roots are the points where the graph intersects the x-axis, and the vertex is the point where the graph changes direction.
Q: How do I find the x-intercepts of a quadratic equation?
A: To find the x-intercepts of a quadratic equation, you can set y = 0 and solve for x. This will give you the x-coordinates of the points where the graph intersects the x-axis.
Q: How do I find the vertex of a quadratic equation?
A: To find the vertex of a quadratic equation, you can use the formula x = -b / 2a. This will give you the x-coordinate of the vertex, and you can substitute this value into the equation to find the y-coordinate.
Q: What is the difference between a quadratic equation and a linear equation?
A: A quadratic equation is a polynomial equation of degree 2, while a linear equation is a polynomial equation of degree 1. Quadratic equations have a parabolic shape, while linear equations have a straight line shape.
Q: How do I graph a quadratic equation with a negative leading coefficient?
A: To graph a quadratic equation with a negative leading coefficient, you can use the same steps as before. However, the graph will open downwards instead of upwards.
Q: What are some common mistakes to avoid when graphing a quadratic equation?
A: Some common mistakes to avoid when graphing a quadratic equation include:
- Not finding the roots and vertex of the equation
- Not plotting the points on the graph correctly
- Not drawing a smooth curve through the points
- Not using the correct formula to find the vertex
Conclusion
Graphing a quadratic equation requires a clear understanding of the equation's properties. By following the steps outlined in this article, you can create an accurate representation of the equation. Remember to find the roots and vertex of the equation, plot the points on the graph, and draw a smooth curve through the points.
Additional Resources
For more information on graphing quadratic equations, check out the following resources:
- Khan Academy: Graphing Quadratic Equations
- Mathway: Graphing Quadratic Equations
- Wolfram Alpha: Graphing Quadratic Equations
Frequently Asked Questions
Q: What is the equation of the graph? A: The equation of the graph is y = -x^2 + 14x.
Q: What are the roots of the equation? A: The roots of the equation are x = 0 and x = 14.
Q: What is the vertex of the graph? A: The vertex of the graph is at x = -7 and y = -49.
Q: How do I graph a quadratic equation? A: To graph a quadratic equation, you can use a graphing tool or software, or you can plot the points on a graph by hand.
Q: What is the significance of the roots and vertex in graphing a quadratic equation? A: The roots and vertex are important features of a quadratic equation that help you understand its graph.
Q: How do I find the x-intercepts of a quadratic equation? A: To find the x-intercepts of a quadratic equation, you can set y = 0 and solve for x.
Q: How do I find the vertex of a quadratic equation? A: To find the vertex of a quadratic equation, you can use the formula x = -b / 2a.
Q: What is the difference between a quadratic equation and a linear equation? A: A quadratic equation is a polynomial equation of degree 2, while a linear equation is a polynomial equation of degree 1.
Q: How do I graph a quadratic equation with a negative leading coefficient? A: To graph a quadratic equation with a negative leading coefficient, you can use the same steps as before. However, the graph will open downwards instead of upwards.
Q: What are some common mistakes to avoid when graphing a quadratic equation? A: Some common mistakes to avoid when graphing a quadratic equation include not finding the roots and vertex of the equation, not plotting the points on the graph correctly, not drawing a smooth curve through the points, and not using the correct formula to find the vertex.