Problem 23 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=6 x^{2} $$

Step-by-Step Solution

Answer

a. The graph opens up. b. The vertex is at the coordinate (0,0). c. The equation of the axis of symmetry is \(x = 0\).

Step 1: Determine the Opening Direction of the Graph

The coefficient in front of the \(x^2\) term is the leading coefficient. If it is positive, the graph will open upwards, if it is negative, it will open downwards. In this case, the leading coefficient is 6, which is positive, indicating that the graph opens up.

Step 2: Find the Coordinates of the Vertex

For a quadratic function in the form \(y = ax^2 + bx + c\), the vertex is given by the coordinates \((-b/2a, f(-b/2a))\). However, in this case, the equation is simplified to \(y = ax^2\), meaning the vertex is (0,0).

Step 3: Write Equation of the Axis of Symmetry

The axis of symmetry for a function in standard form \(y = ax^2 + bx + c\) is given by the equation \(x = -b/2a\). Here, since our function is \(y = 6x^2\), the axis of symmetry will be \(x = 0\).

Key Concepts

Graph of Quadratic Function

A quadratic function is represented on a graph as a parabola, which is a smooth U-shaped or inverted U-shaped curve. The basic form of a quadratic function is expressed as y = ax^2 + bx + c, where a, b, and c are constants, and x and y are the variables.

The value of the constant a is particularly important because it influences the direction in which the parabola opens. If a is positive, the parabola opens upward, creating the familiar U-shape. Conversely, if a is negative, the parabola opens downward, forming an inverted U-shape.

For the given function y=6x^2, the constant a is 6, which is a positive value. This tells us that the graph of this quadratic function will open upwards. The graph's width and how sharply it bends depend on the magnitude of a. A larger magnitude results in a narrower parabola, while a smaller one leads to a wider curve.

Vertex of a Parabola

The vertex of a parabola is its highest or lowest point, depending on whether it opens up or down, and it is a crucial feature of the graph. For the standard form of a quadratic function y = ax^2 + bx + c, the vertex can be found using the formula (-b/2a, f(-b/2a)).

However, when a quadratic function is in the form y = ax^2, which lacks the bx and c terms, the calculation simplifies considerably. Since there is no b term in the given function y=6x^2, the coordinates of the vertex are (0,0). This central point is where the parabola either reaches its minimum value, if it opens upwards, or its maximum value, if it opens downwards.

In problems involving quadratic functions, identifying the vertex provides key information about the shape and position of the parabola on the graph, and it helps in sketching the curve accurately.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirrored halves. It passes through the vertex and represents a fundamental symmetry in the graph of a quadratic function. For the standard quadratic equation y = ax^2 + bx + c, the equation of the axis of symmetry can be found using the formula x = -b/2a.

With the quadratic expression y=6x^2, the absence of the b term simplifies the calculation of the axis of symmetry, which becomes x = 0. This tells us that the axis of symmetry is the y-axis itself. Any point on the parabola has a corresponding point on the opposite side of this line, at an equal distance from it.

The concept of the axis of symmetry is especially valuable when graphing a quadratic function, as it ensures the parabola is accurately represented in terms of balance and alignment. It also assists in determining the domain and range of the function, as it clearly marks the fold line of the parabola's graph.