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Magazine Chapter 9: Problem 47
Find the vertex of the parabola. $$ y=x(x-1) $$
Short Answer
Expert verified
Answer: The vertex of the parabola is $$\left(\frac{1}{2}, -\frac{1}{4}\right)$$.
Step by step solution
01
Rewrite the parabola equation in the general quadratic form
To rewrite the given equation in the general quadratic form, we need to expand it:
$$
y = x(x - 1) = x^2 - x
$$
Now, the parabola is in the general quadratic form, which is given by $$y = ax^2 + bx + c$$.
02
Complete the square
In order to write the equation in the vertex form, we need to complete the square:
$$
y = x^2 - x
$$
To complete the square, we need to add and subtract the term $$(\frac{b}{2a})^2$$ to the equation, where a = 1 and b = -1.
$$
(\frac{-1}{2(1)})^2 = (\frac{1}{2})^2 = \frac{1}{4}
$$
So we will add and subtract the term $$\frac{1}{4}$$ to the equation:
$$
y = (x^2 - x + \frac{1}{4}) - \frac{1}{4}
$$
Now, the equation inside the parentheses is a perfect square trinomial that can be factored into a binomial square:
$$
y = (x - \frac{1}{2})^2 - \frac{1}{4}
$$
03
Identify the vertex
The vertex form of the parabola is given by $$y = a(x - h)^2 + k$$. In our case, this takes the form:
$$
y = (x - \frac{1}{2})^2 - \frac{1}{4}
$$
Comparing it with the vertex form, we can see that the vertex (h, k) is given by:
$$
h = \frac{1}{2} \ \text{and} \ k = -\frac{1}{4}
$$
So the vertex of the parabola is $$\left(\frac{1}{2}, -\frac{1}{4}\right)$$.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a algebraic method used to transform a quadratic expression into a perfect square trinomial. This process is particularly valuable because it allows us to rewrite quadratic equations in vertex form, making it easier to identify the vertex of a parabola. Here's a quick guide on how to complete the square for a quadratic equation:
- Begin with the standard form of a quadratic equation: \( ax^2 + bx + c \).
- Divide the coefficient of \( x \) (which is \( b \)) by 2, and then square the result to find \( \left( \frac{b}{2a} \right)^2 \).
- Add this square inside the equation to complete the square, and immediately subtract it as well to maintain balance, resulting in \( (x + m)^2 - n \), where \( m = \frac{b}{2a} \) and \( n = \left( \frac{b}{2a} \right)^2 \).
- Your equation is now ready to be reframed in vertex form: \( a(x-h)^2 + k \), where the vertex is \( (h, k) \).
Quadratic Equation
The quadratic equation is a mathematical statement expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This equation is central to algebra because it represents parabolas graphically.The solutions of a quadratic equation are found using various methods, such as factoring, taking square roots, and using the quadratic formula:
- The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), provides a way to find the roots (or solutions) of any quadratic equation.
- These roots can be graphically interpreted as the points where the parabola intersects the x-axis.
- The discriminant \( b^2 - 4ac \) determines the nature of the roots; if positive, the parabola intersects the x-axis at two distinct points. If zero, there's exactly one point of intersection (vertex), and if negative, the parabola does not intersect the x-axis at all.
Parabola
A parabola is a U-shaped curve that results from graphing a quadratic function. Formally, it is defined by the equation \( y = ax^2 + bx + c \) where \( a, b, \) and \( c \) are constants and \( a eq 0 \). Understanding a parabola involves several key properties:
- **Vertex**: The highest or lowest point on a parabola, which gives indication of the parabola's maximum or minimum value, depending on its orientation.
- **Axis of Symmetry**: A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It can be found using \( x = -\frac{b}{2a} \).
- **Direction**: If the coefficient \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.
- **Focus and Directrix**: These are geometric components that further define the parabola鈥檚 orientation and shape; the parabola is the set of all points equidistant from the focus and the directrix line.
