Problem 24 - Algebra 2

Question

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=x^{2}-6 x+1 $$

Step-by-Step Solution

Answer

Vertex: (3, -8); Axis of symmetry: x = 3; Opens upwards.

Step 1: Start with the Original Quadratic Function

The given quadratic function is \( y = x^2 - 6x + 1 \).

Step 2: Rewrite in Vertex Form

A quadratic function in vertex form is \( y = a(x - h)^2 + k \). We need to complete the square to transform the given function into this form.

Step 3: Complete the Square

Identify the coefficient of \( x \), which is \(-6\). Take half of it, square it, then add and subtract this number inside the equation to complete the square.- Half of \(-6\) is \(-3\).- Squaring \(-3\) gives \(9\).We rewrite the original equation: \( y = x^2 - 6x + 9 - 9 + 1 \).

Step 4: Simplify the Equation

This becomes:\( y = (x-3)^2 - 8 \).This is the vertex form \( y = a(x-h)^2 + k \), where \( a = 1 \), \( h = 3 \), and \( k = -8 \).

Step 5: Identify the Vertex

The vertex \((h, k)\) is \((3, -8)\).

Step 6: Identify the Axis of Symmetry

The axis of symmetry is the vertical line \( x = h \). Thus, it is \( x = 3 \).

Step 7: Determine the Direction of Opening

Since the coefficient \( a \) (which is 1) is positive, the parabola opens upwards.

Key Concepts

Vertex Form

In the world of quadratic functions, the vertex form plays a significant role in simplifying the analysis of parabolas. The vertex form of a quadratic function is expressed as \( y = a(x-h)^2 + k \). This form is ideal for easily identifying the vertex of the parabola, which is the point \((h, k)\).
This format is crucial because it provides quick insights into the characteristics of the graph, such as its position and direction. For example, if we have \( y = (x-3)^2 - 8 \), it's in vertex form, revealing the vertex at \((3, -8)\). This enables us to pinpoint where the graph has its peak or trough on the coordinate plane, depending on the direction in which the parabola opens.

Completing the Square

Completing the square is a method used to transform a standard quadratic function into vertex form. It involves manipulating the original quadratic to form a perfect square trinomial.

Here's how it's done:

• Take the quadratic function, such as \( y = x^2 - 6x + 1 \).
• Identify the linear coefficient, which is \(-6\), and take half of it, resulting in \(-3\).
• Square \(-3\) to get \(9\).
• Add and subtract \(9\) within the function, i.e., \( y = x^2 - 6x + 9 - 9 + 1 \).

Simplify and rearrange to form \( y = (x-3)^2 - 8 \).
This transformation makes the quadratic much easier to work with, especially when identifying other features like the vertex and the axis of symmetry.

Axis of Symmetry

The axis of symmetry in a quadratic function is a line that vertically splits the parabola into two mirror-image halves. In vertex form, \( y = a(x-h)^2 + k \), the axis of symmetry is clearly identified as \( x = h \).
This specific line crosses the parabola at its vertex.

• For the function \( y = (x-3)^2 - 8 \), the axis of symmetry is the line \( x = 3 \).

Knowing this line is beneficial because it represents the point of balance on the graph, where one side is a mirror image of the other. It also often helps in graphing, as a reference for plotting points and understanding the shape of the function.

Direction of Opening

The direction of opening for a quadratic function refers to whether the parabola opens upwards or downwards. This is determined by the sign of the coefficient \( a \) in the vertex form \( y = a(x-h)^2 + k \).

• If \( a > 0 \), the parabola opens upwards, resembling a 'U' shape.
• If \( a < 0 \), the parabola opens downwards, similar to an upside-down 'U'.

In our case with \( y = (x-3)^2 - 8 \), \( a = 1 \), which is positive, indicating that the parabola opens upwards.
Understanding the direction of opening helps in predicting the behavior of the parabola and its vertex. It can also aid in determining the range of the function and the nature of the quadratic's maximum or minimum value.