Problem 52 - Algebra 2

Question

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=x^{2}-2 x+9 $$

Step-by-Step Solution

Answer

The vertex form is \( y = (x-1)^2 + 8 \). The vertex is \((1, 8)\), the axis of symmetry is \(x = 1\), and the parabola opens upwards.

Step 1: Identify the standard quadratic form

The given equation is \( y = x^2 - 2x + 9 \). This is in the standard quadratic form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -2 \), and \( c = 9 \). Our task is to convert this into the vertex form, which is \( y = a(x-h)^2 + k \).

Step 2: Complete the square

To convert the equation into vertex form, we need to complete the square for the quadratic expression \( x^2 - 2x \). Begin by taking the coefficient of \( x \), which is \( -2 \), divide it by 2 to get \( -1 \), then square it to get \( 1 \). Add and subtract this inside the equation: \( y = (x^2 - 2x + 1) - 1 + 9 \).

Step 3: Rearrange the expression

The expression \( x^2 - 2x + 1 \) can be rewritten as \( (x-1)^2 \). By doing this, our equation becomes \( y = (x-1)^2 - 1 + 9 \).

Step 4: Simplify the equation

Combine the constant terms: \( -1 + 9 = 8 \). Therefore, the equation in vertex form is \( y = (x-1)^2 + 8 \).

Step 5: Identify the vertex and axis of symmetry

In the vertex form \( y = a(x-h)^2 + k \), the vertex is \((h, k)\). Therefore, for \( y = (x-1)^2 + 8 \), the vertex is \((1, 8)\). The axis of symmetry is the line \( x = h \), so it is \( x = 1 \) here.

Step 6: Determine the direction of opening

Since \( a = 1 \) (the coefficient of the \( (x-h)^2 \) term), which is positive, the parabola opens upwards.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic equations. It helps in transforming an equation from the standard quadratic form to the vertex form. The standard quadratic form is expressed as \( y = ax^2 + bx + c \). To convert this into the vertex form, \( y = a(x-h)^2 + k \), we "complete the square."

To complete the square, start with the quadratic term and the linear term in the equation. For example, in \( y = x^2 - 2x + 9 \), identify the parts \( x^2 \) and \( -2x \). Take the coefficient of \( x \), which is \(-2\), divide it by 2, getting \(-1\), and square it, resulting in \(1\). Add and subtract this number inside the expression to maintain equality. So the equation becomes \( y = (x^2 - 2x + 1) - 1 + 9 \).

By rearranging and simplifying it, \( x^2 - 2x + 1 \) turns into \( (x-1)^2 \). Completing the square is crucial because it sets up the equation to easily reveal the nature of the quadratic curve.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they include terms up to \( x^2 \). They are typically written in the standard form \( y = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) represent constant coefficients.

The equation \( y = x^2 - 2x + 9 \) is in this form where \( a = 1 \), \( b = -2 \), and \( c = 9 \). Quadratic equations can be represented graphically as parabolas. Understanding and transforming these equations into vertex form \( y = a(x-h)^2 + k \) helps in analyzing their geometric properties, such as position and orientation.

Solving quadratic equations is a fundamental skill that encompasses various techniques like factoring, using the quadratic formula, or completing the square, each providing insight into the equation's roots and graph shape.

Parabola Characteristics

The quadratic equation in vertex form helps us understand the characteristics of a parabola, which is the curve formed by the graph of the equation.

• The vertex \((h, k)\) is the peak or trough of the parabola.
• The coefficient \(a\) determines whether the parabola opens upwards or downwards. A positive \(a\) means it opens upwards, while a negative \(a\) means downwards.
• The vertex form equation \( y = (x-1)^2 + 8 \) indicates the vertex at \((1, 8)\).

This form also allows us to easily see that the parabola opens upwards since \( a = 1 \), clearly indicating an upward direction. The vertex form clearly shows the position and shape of the parabola, making it an effective tool to analyze its characteristics.

Axis of Symmetry

The axis of symmetry is an important line in the analysis of parabolas, acting as a mirror line that divides the parabola into two identical halves. It runs vertically through the vertex. In vertex form \( y = a(x-h)^2 + k \), the axis of symmetry is the line \( x = h \).

For the equation \( y = (x-1)^2 + 8 \), the axis of symmetry is \( x = 1 \). This line indicates the vertical path on the coordinate plane where the parabola reflects itself on both sides. Understanding the axis of symmetry helps in predicting the behavior and plotting the graph of a quadratic function more effectively.

Identifying this axis is crucial in problems involving symmetry and optimizations involving quadratic functions, providing a reference line that relates directly to the key features of the parabola.