When faced with a quadratic equation, the most effective approach is to visualize the set of solutions it represents. By looking at mathematical concepts graphically, we can easily identify patterns and analyze trends. This is especially true for quadratic equations, which can be represented by a parabola.
Before we dive into graphing, let's first review what a quadratic equation is. It is a polynomial equation in the form y = ax2 + bx + c, where a ≠ 0. This can also be expressed as a function, y = f(x) ⇔ f(x) = ax2 + bx + c. Here, ax2 is the quadratic term, bx is the linear term, and c is the constant term.
Graphing quadratic equations is a useful method for finding solutions and identifying key behaviors within the equation. The graph of a quadratic equation has a special name: a parabola. Throughout this topic, we will explore different techniques for plotting these equations. But first, let's understand the components of a parabola.
The graph of a Quadratic Equation
A parabola has an axis of symmetry, which can be found using the formula . This line divides the parabola into two equal halves. The point where the axis of symmetry intersects the parabola is known as the vertex. The y-intercept can be found by plugging x = 0 into the quadratic equation, yielding y = c. The x-intercept can be found by setting the quadratic equation to zero: ax2 + bx + c = 0. The x-coordinate of the vertex is and the y-coordinate is .
A parabola's vertex can represent a maximum or minimum value, making it the turning point of the curve. This is where the graph changes direction, transitioning from increasing to decreasing or vice versa. The maximum value is the highest point on the curve, while the minimum value is the lowest. The coefficient a in the quadratic equation y = ax2 + bx + c determines the nature of the turning point. Let's look at two cases: a > 0 and a < 0.
The solutions of a quadratic equation, also known as roots or zeros, are represented by the x-intercepts of the parabola. They can be found by setting y to zero and solving for x in the quadratic equation ax2 + bx + c = 0. But how do we determine the number of solutions a quadratic equation might have? This is where the discriminant comes into play. It can either have one real solution, two real solutions, or no real solutions, depending on the sign of the discriminant.
The discriminant, represented by D = b2 – 4ac, helps us identify the number and type of roots of a quadratic equation. For a quadratic equation ax2 + bx + c = 0, where a ≠ 0, there are three cases to consider. Let's take a look at them and how they are represented graphically.
This method involves plotting the quadratic equation and creating a table of values. Here's how it works:
In this step, we will apply the Location Principle to estimate the solutions of a given quadratic equation. This principle, along with three examples, will be explained below.
The x-intercept of a graph, where y = 0, changes the sign of the y-values. This means that the Location Principle is looking for a change in sign between two outputs of y given two inputs of x.
Consider the function f(x) = 2x2 – 4x – 3. We need to determine whether the function has a maximum or minimum value and then find its corresponding value. The domain and range of the function will also be stated.
When solving quadratic equations, it's helpful to be familiar with the standard and intercept forms, as well as different methods of factoring. Let's explore these concepts using examples.
Example 1: Given the equation y = 2x2 - 4x - 3, let's determine the vertex, minimum value, domain, and range.
Solution: We can identify that a = 2, b = -4, and c = -3. Since a > 0, the parabola opens upwards and the function has a minimum value. The minimum value is the y-coordinate of the vertex, which can be found using the formula. This gives us an x-coordinate of 1 and a y-coordinate of -5, making the vertex (1, -5). The domain is all real numbers, while the range is all real numbers equal to or greater than -5. This can also be expressed in set notation as [–5, +∞ [.
The graph of this function is displayed below:
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