Problem 25 - 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. $$ y=5 x^{2}-6 $$

Step-by-Step Solution

Answer

Vertex: (0, -6), Axis of symmetry: x = 0, Opens upwards.

Step 1: Convert to Vertex Form

The vertex form of a quadratic equation is given as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. First, recognize that the given function \( y = 5x^2 - 6 \) is in standard form \( ax^2 + bx + c \), with \( b = 0 \). Rewrite it by completing the square, essentially since \( b = 0 \), the term inside the square remains only \( x^2 \). Thus, it is already almost in the vertex form \( 5(x - 0)^2 - 6 \).

Step 2: Identify the Vertex

The equation in vertex form \( y = 5(x - 0)^2 - 6 \) indicates that the vertex \((h, k)\) is \((0, -6)\). This is because there is no horizontal shift (\(h = 0\)), and the parabola is vertically shifted down by \(6\) units.

Step 3: Axis of Symmetry

The axis of symmetry for a parabola in the form \( f(x) = a(x-h)^2 + k \) is a vertical line given by \( x = h \). Here, since \( h = 0 \), the axis of symmetry is the line \( x = 0 \).

Step 4: Direction of Opening

Check the coefficient of \( (x-h)^2 \) which is \( a = 5 \) in this case. Since \( a > 0 \), the parabola opens upwards.

Key Concepts

complete the square

Completing the square is a method used to rewrite a quadratic equation in vertex form. This technique helps to identify the vertex of the corresponding parabola easily. In the given exercise, the quadratic function is \[ y = 5x^2 - 6 \]To convert it into vertex form, which is \[ y = a(x-h)^2 + k, \]we normally transform the expression \[ ax^2 + bx + c \]by manipulating it into a perfect square trinomial. However, since the term \( b \) in our equation is zero, the expression is already quite simplified.

• The expression \( 5x^2 \) can be rewritten as \( 5(x - 0)^2 \) because \( b = 0 \) leads to no additional term needed inside the bracket.
• Hence, the equation becomes \[ y = 5(x-0)^2 - 6, \]
• giving a clear vertex form of the equation.

Completing the square simplifies the process to determine the attributes of a parabola like its vertex, symmetry, and direction.

axis of symmetry

The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. For a quadratic equation in vertex form \[ y = a(x-h)^2 + k, \]this line is represented by \[ x = h. \]In our quadratic, \[ y = 5(x-0)^2 - 6, \]we see that \( h = 0 \).

• This makes the axis of symmetry the line \( x = 0 \).
• Being a straight line, it crosses the x-axis at the vertex of the parabola.

This axis of symmetry is crucial because it helps in understanding the balance and structure of the graph, ensuring calculations are performed uniformly on either side of this line.

parabola

A parabola is the graph of a quadratic function. It is a U-shaped curve, and its shape and orientation provide important insights into the equation. For the function \[ y = 5x^2 - 6, \]the parabola has certain distinguishable features:

• The vertex is at the point \( (0, -6) \), clearly marking the lowest point of this parabola.
• Being symmetric about the line \( x = 0 \), the curve mirrors itself perfectly across this line.

The role of the vertex in the parabola is pivotal since it defines the graph's extremity (either the lowest or highest point, based on the direction of opening). Identifying this allows for deeper analysis of the graph's behavior.

direction of opening

The direction of opening of a parabola tells us which way the curve faces. In the vertex form \[ y = a(x-h)^2 + k, \]the sign of the coefficient \( a \) dictates this direction.

• If \( a > 0 \), the parabola opens upwards, forming a U-shape.
• If \( a < 0 \), it opens downwards, creating an inverted U-shape.

For our function \[ y = 5(x-0)^2 - 6, \]the \( a \) value is 5, which is positive.

• This means the parabola opens upwards.

This characteristic helps in understanding the graph's orientation and is essential for predicting the curve's behavior across different values of \( x \). Understanding the direction aids in visualizing and sketching the graph accurately.