Problem 5 - 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. $$ y=5(x+3)^{2}-1 $$

Step-by-Step Solution

Answer

Vertex: (-3, -1); Axis: x = -3; Opens upwards.

Step 1: Verify the Existing Form

The quadratic function given is \( y = 5(x+3)^2 - 1 \). Notice that it follows the vertex form \( y = a(x-h)^2 + k \).

Step 2: Identify the Vertex

In the vertex form \( y = a(x-h)^2 + k \), the vertex is \((h, k)\). Here, \(h = -3\) and \(k = -1\). The vertex is \((-3, -1)\).

Step 3: Identify the Axis of Symmetry

The axis of symmetry for the quadratic equation in vertex form is \( x = h \). Hence, the axis of symmetry is \( x = -3 \).

Step 4: Determine the Direction of Opening

The direction of opening is determined by the sign of \( a \) in the vertex form \( y = a(x-h)^2 + k \). Since \( a = 5 \), the parabola opens upwards.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be written in the form of a second-degree equation. Its general form is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). However, quadratic functions are often expressed in vertex form as well, which is \( y = a(x-h)^2 + k \). This expression makes it easier to identify key features of the graph such as the vertex, axis of symmetry, and direction of opening. The vertex form is particularly useful because it provides direct insight into the shape and position of the parabola represented by the quadratic function. Understanding the vertex form can greatly aid in graphing and analyzing quadratic functions, as well as solving related problems.

Vertex

The vertex of a quadratic function is a critical point that could represent either the minimum or maximum point on the graph, depending on the direction the parabola opens. In the vertex form equation \( y = a(x-h)^2 + k \), the vertex is given by the coordinates \((h, k)\). This means you can directly read off the vertex from the equation without additional calculations. For the given quadratic function \( y = 5(x+3)^2 - 1 \):- Compare it with the vertex form \( y = a(x-h)^2+k \).- The vertex is found at \( h = -3 \) and \( k = -1 \).Therefore, the vertex is at \((-3, -1)\). The vertex is a particularly important feature for graphing, as it marks the point of peak or trough of the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It can be thought of as a mirror line, reflecting each half of the parabola across it.For quadratic functions in vertex form, the axis of symmetry can be found using the equation \( x = h \). This aligns with the \( h \) value of the vertex.In our equation \( y = 5(x+3)^2 - 1 \):- The vertex \( h \) value is \(-3\).- Thus, the axis of symmetry is \( x = -3 \).This axis of symmetry is useful when sketching the parabola because it indicates where the parabolic graph will be balanced, showing equal behavior on the left and right of this line.

Direction of Opening

The direction in which the parabola opens is determined by the value of \( a \) in the vertex form of a quadratic function. Whether the parabola opens upwards or downwards depends solely on whether \( a \) is positive or negative:- If \( a > 0 \), the parabola opens upwards.- If \( a < 0 \), the parabola opens downwards.In our specific function \( y = 5(x+3)^2 - 1 \), the \( a \) value is \( 5 \), which is positive. This indicates that the parabola opens upwards. This orientation tells us that the vertex is the minimum point of the graph. Understanding the direction of opening is crucial for predicting how the function behaves, particularly as \( x \) moves away from the vertex.