Problem 31 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$y=x^{2}-4 x+1$$

Step-by-Step Solution

Answer

The vertex of the quadratic function is (2, -3).

Step 1: Identify the quadratic function form

The given quadratic function is of the form \(y = ax^2 + bx + c\). Here, the coefficients are:\(a = 1\), \(b = -4\), and \(c = 1\).

Step 2: Use the vertex formula for x-coordinate

For a quadratic function in the form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula: \(x = -\frac{b}{2a}\).Substitute the values of \(a\) and \(b\) into the formula:\(x = -\frac{-4}{2(1)} = \frac{4}{2} = 2\).

Step 3: Find the y-coordinate of the vertex

To find the y-coordinate, substitute the x-coordinate of the vertex back into the original quadratic equation. Given \(x = 2\), substitute it into \(y = x^2 - 4x + 1\):\(y = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3\).

Key Concepts

quadratic equations

Quadratic equations form the cornerstone of many algebraic concepts and have a general form expressed as:
\(y = ax^2 + bx + c\).
These equations always graph into a U-shaped curve known as a parabola.
In a quadratic equation, the highest exponent of the variable, usually \(x\), is 2. This makes them different from linear equations, where the highest exponent is 1.
Quadratic equations can be solved using various methods such as:

• Factoring
• Completing the square
• Using the quadratic formula

vertex formula

The vertex of a quadratic function is a key component. It is the highest or lowest point on the graph of the function, depending on the parabola's orientation.
To find the vertex, we use the vertex formula:
\( x = -\frac{b}{2a} \).
Here, \(x\) is the x-coordinate of the vertex, and \(a\) and \(b\) are coefficients from the quadratic equation \(y = ax^2 + bx + c\).
Once we find the x-coordinate, we can substitute it back into the original equation to find the y-coordinate.
This way, you can precisely locate the vertex of your quadratic function.

parabola properties

Parabolas are unique curves derived from quadratic equations.
Several properties define a parabola, including:

• Vertex: The peak or the lowest point on the graph.
• Axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• Direction: If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• Focus and Directrix: Special lines associated with the parabolic curve, which help in geometric interpretations.

Understanding these properties is crucial for drawing and analyzing quadratic functions.

standard form of quadratic function

The standard form of a quadratic function is written as:
\(y = ax^2 + bx + c\).
Here, \(a\), \(b\), and \(c\) are constants.
This form allows for easy application of the vertex formula and other methods of solving quadratic equations.
For example, in the equation \(y = x^2 - 4x + 1\), we identified:

• \(a = 1\)
• \(b = -4\)
• \(c = 1\)

Using these coefficients, we determined the vertex to understand the graph's peak accurately. The standard form is a foundational aspect of working with quadratic equations.