Graphing Quadratic Equations

In the previous section, we learned about the basics of graphing and how to plot graphs using the general rules and principles of graphical construction. In this section, we will focus on graphing quadratic equations.

A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve.

To graph a quadratic equation, we need to identify the key components of the equation: the vertex, the axis of symmetry, and the x-intercepts.

Finding the Vertex

The vertex of a quadratic equation represents the minimum or maximum point on the graph. To find the vertex, we can use the formula x = -b/2a. Let’s consider an example:

Example:

Graph the quadratic equation y = x^2 – 4x + 3.

To find the vertex, we can use the formula x = -b/2a.

In this case, a = 1 and b = -4. Plugging these values into the formula, we get:

x = -(-4)/(2*1) = 4/2 = 2

So, the x-coordinate of the vertex is 2. To find the y-coordinate, we substitute the x-coordinate into the equation:

y = (2)^2 – 4(2) + 3 = 4 – 8 + 3 = -1

Therefore, the vertex of the quadratic equation y = x^2 – 4x + 3 is (2, -1).

Finding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The formula for the axis of symmetry is x = -b/2a. Using the same example as before, the axis of symmetry is x = 2.

Finding the x-intercepts

The x-intercepts, also known as the roots or zeros, are the points where the graph intersects the x-axis. To find the x-intercepts, we set y = 0 and solve the quadratic equation. Let’s continue with the same example:

y = x^2 – 4x + 3

Setting y = 0:

0 = x^2 – 4x + 3

Factoring the equation, we get:

0 = (x – 1)(x – 3)

Solving for x, we find:

x – 1 = 0 or x – 3 = 0

x = 1 or x = 3

Therefore, the x-intercepts of the quadratic equation y = x^2 – 4x + 3 are x = 1 and x = 3.

Graphing the Quadratic Equation

Now that we have all the necessary information, we can plot the graph of the quadratic equation y = x^2 – 4x + 3.

First, we plot the vertex, which is (2, -1). Then, we draw the axis of symmetry, which is x = 2. Next, we plot the x-intercepts, which are (1, 0) and (3, 0). Finally, we draw the parabola, making sure it passes through the vertex and the x-intercepts.

By following these steps, we can accurately graph any quadratic equation.

Conclusion

In this section, we learned about graphing quadratic equations. We explored how to find the vertex, the axis of symmetry, and the x-intercepts. By understanding these key components, we can accurately graph quadratic equations. Graphing is an essential skill in accounting, as it allows us to visually represent data and make informed decisions. In the next section, we will further explore the interpretation of mathematical graphs in the context of accounting data.