Examples of Graphing Quadratic Equations

In this section, we will explore some examples of graphing quadratic equations. Quadratic equations are polynomial equations of degree 2, which can be written in the form:

y = ax^2 + bx + c

where a, b, and c are constants.

Example 1: Graphing a Simple Quadratic Equation

Let’s consider the equation:

y = x^2 – 4x + 4

To graph this equation, we will plot several points and connect them to form a smooth curve.

We can start by finding the y-intercept of the equation. The y-intercept is the value of y when x is equal to 0. Plugging in x = 0 into the equation, we get:

y = (0)^2 – 4(0) + 4 = 4

So, the y-intercept is (0, 4).

Next, let’s find the x-intercepts of the equation. The x-intercepts are the values of x when y is equal to 0. To find the x-intercepts, we need to solve the equation:

x^2 – 4x + 4 = 0

This equation can be factored as:

(x – 2)(x – 2) = 0

So, the equation has a double root at x = 2. Therefore, the x-intercept is (2, 0).

Now, let’s find a few more points to plot on the graph. We can choose some values for x and calculate the corresponding values of y. For example, when x = 1:

y = (1)^2 – 4(1) + 4 = 1 – 4 + 4 = 1

So, we have the point (1, 1).

We can repeat this process for a few more values of x to get more points. Once we have enough points, we can plot them on a graph and connect them to form a smooth curve.

After plotting all the points, we can see that the graph of the equation is a parabola that opens upwards. The vertex of the parabola is located at the point (2, 0).

Example 2: Graphing a Quadratic Equation with a Negative Leading Coefficient

Let’s consider the equation:

y = -2x^2 + 4x – 2

Similar to the previous example, we will plot points and connect them to graph this equation.

First, let’s find the y-intercept by plugging in x = 0:

y = -2(0)^2 + 4(0) – 2 = -2

So, the y-intercept is (0, -2).

Next, let’s find the x-intercepts by solving the equation:

-2x^2 + 4x – 2 = 0

This equation can be factored as:

-2(x – 1)(x – 1) = 0

So, the equation has a double root at x = 1. Therefore, the x-intercept is (1, 0).

Now, let’s find a few more points to plot on the graph. For example, when x = -1:

y = -2(-1)^2 + 4(-1) – 2 = -2 + 4 – 2 = 0

So, we have the point (-1, 0).

By plotting all the points and connecting them, we can see that the graph of the equation is a parabola that opens downwards. The vertex of the parabola is located at the point (1, -2).

Conclusion

Graphing quadratic equations allows us to visualize the relationship between the dependent variable (y) and the independent variable (x). By plotting points and connecting them, we can identify key features of the graph, such as the vertex, intercepts, and the direction in which the parabola opens.

Understanding how to graph quadratic equations is essential for analysing and interpreting mathematical graphs in the field of accounting. It enables us to make informed decisions based on graphical data and identify any misrepresentations or inaccuracies in the data.

In the next section, we will explore the concept of identifying dependent and independent variables in mathematical graphs and how they relate to accounting data.