Graph The Equation $y = -x^2 + 8x - 12$ On The Accompanying Set Of Axes. You Must Plot 5 Points, Including The Roots And The Vertex.

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Introduction

Graphing quadratic equations is an essential skill in mathematics, particularly in algebra and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on graphing the equation $y = -x^2 + 8x - 12$ on a set of axes. We will also identify the roots and the vertex of the parabola.

Understanding Quadratic Equations

A quadratic equation has the general form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$. If $a > 0$, the parabola opens upwards, and if $a < 0$, the parabola opens downwards.

Graphing the Equation

To graph the equation $y = -x^2 + 8x - 12$, we need to find the roots and the vertex of the parabola. The roots of the parabola are the x-coordinates of the points where the parabola intersects the x-axis. The vertex of the parabola is the highest or lowest point on the parabola.

Finding the Roots

To find the roots of the parabola, we need to set the equation equal to zero and solve for $x$. We can do this by factoring the equation or by using the quadratic formula.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the equation
equation = -x**2 + 8*x - 12

# Solve the equation
roots = sp.solve(equation, x)

print(roots)

The roots of the parabola are $x = 3$ and $x = 4$.

Finding the Vertex

To find the vertex of the parabola, we need to use the formula $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = 8$.

# Define the variables
a = -1
b = 8

# Calculate the x-coordinate of the vertex
x_vertex = -b / (2 * a)

print(x_vertex)

The x-coordinate of the vertex is $x = -4$. To find the y-coordinate of the vertex, we need to plug this value into the equation.

# Define the equation
equation = -x**2 + 8*x - 12

# Substitute the x-coordinate of the vertex into the equation
y_vertex = equation.subs(x, x_vertex)

print(y_vertex)

The y-coordinate of the vertex is $y = 16$.

Plotting the Points

Now that we have found the roots and the vertex of the parabola, we can plot the points on the graph.

x y
3 0
4 0
-4 16

Graphing the Parabola

To graph the parabola, we can use a graphing tool or software. In this case, we will use a simple graphing tool to plot the points and draw the parabola.

Conclusion

Graphing quadratic equations is an essential skill in mathematics. In this article, we have graphed the equation $y = -x^2 + 8x - 12$ on a set of axes. We have identified the roots and the vertex of the parabola and plotted the points on the graph. We have also used a simple graphing tool to draw the parabola.

Tips and Variations

  • To graph a quadratic equation, you need to find the roots and the vertex of the parabola.
  • The roots of the parabola are the x-coordinates of the points where the parabola intersects the x-axis.
  • The vertex of the parabola is the highest or lowest point on the parabola.
  • You can use a graphing tool or software to graph the parabola.
  • You can also use a simple graphing tool to plot the points and draw the parabola.

Further Reading

  • For more information on graphing quadratic equations, please refer to the following resources:
  • Khan Academy: Graphing Quadratic Equations
  • Mathway: Graphing Quadratic Equations
  • Wolfram Alpha: Graphing Quadratic Equations

References

Introduction

Graphing quadratic equations is an essential skill in mathematics, particularly in algebra and calculus. In our previous article, we graphed the equation $y = -x^2 + 8x - 12$ on a set of axes and identified the roots and the vertex of the parabola. In this article, we will answer some frequently asked questions about graphing quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It has the general form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the graph of a quadratic equation?

A: The graph of a quadratic equation is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$. If $a > 0$, the parabola opens upwards, and if $a < 0$, the parabola opens downwards.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you need to set the equation equal to zero and solve for $x$. You can do this by factoring the equation or by using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to find the roots of a quadratic equation. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

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

A: To find the vertex of a quadratic equation, you need to use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you need to plug this value into the equation.

Q: What is the significance of the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the highest or lowest point on the parabola. It is a critical point that determines the direction of the parabola.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to find the roots and the vertex of the parabola. You can then plot the points on the graph and draw the parabola.

Q: What are some common mistakes to avoid when graphing quadratic equations?

A: Some common mistakes to avoid when graphing quadratic equations include:

  • Not finding the roots and the vertex of the parabola
  • Not plotting the points on the graph correctly
  • Not drawing the parabola correctly
  • Not using the correct formula for the quadratic equation

Conclusion

Graphing quadratic equations is an essential skill in mathematics. In this article, we have answered some frequently asked questions about graphing quadratic equations. We hope that this article has been helpful in clarifying any doubts you may have had about graphing quadratic equations.

Tips and Variations

  • To graph a quadratic equation, you need to find the roots and the vertex of the parabola.
  • The roots of the parabola are the x-coordinates of the points where the parabola intersects the x-axis.
  • The vertex of the parabola is the highest or lowest point on the parabola.
  • You can use a graphing tool or software to graph the parabola.
  • You can also use a simple graphing tool to plot the points and draw the parabola.

Further Reading

  • For more information on graphing quadratic equations, please refer to the following resources:
  • Khan Academy: Graphing Quadratic Equations
  • Mathway: Graphing Quadratic Equations
  • Wolfram Alpha: Graphing Quadratic Equations

References