Question

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation. \(y=4 x^2\)

Short answer

The focus of the parabola \(y = 4x^2\) is at (0, 1), the directrix is the line y = -1, and the axis of symmetry is the line x = 0.

Step-by-step solution

Identify the constant a

From the equation, we can see that the value of a, in '4a', matches with the coefficient of \(x^2\), which is 4. Hence, a = 1.

Determine the focus

The focus of a parabola with the equation \(y = 4ax^2\) is at (0, a). Thus, as a = 1, the focus of the parabola is at (0, 1).

Identify the directrix

The directrix of the parabola is given by the line y = -a. With a = 1, the directrix of the parabola is the line y = -1.

Ascertain the axis of symmetry

The axis of symmetry of the parabola \(y = 4ax^2\) is the line x = 0. This doesn't change with the value of a.

Sketch the Graph

Start by plotting the focus and drawing the directrix. The parabola's vertex is the origin (0,0). The parabola opens upwards since the coefficient of \(x^2\) is positive. Draw a parabolic curve starting from the vertex, going to the focus, and rounding vertically parallel to and above the directrix.

Key concepts

Focus of Parabola

The focus of a parabola is a point that holds a unique property giving the parabola its distinctive curved shape. The parabola is defined such that each point along its arc is equidistant from the focus and a line called the directrix. In our example, given the equation \(y = 4x^2\), the focus is a point above the vertex.

• For the equation in the form \(y = 4ax^2\), the focus is located at \((0, a)\).
• Since "a" is 1 in our equation, the focus for our parabola is at the point \((0, 1)\).

The focus’s position helps in understanding how the parabola curves around this point,acting as a 'guide' around which the curve is formed. Whether the parabola opens upwards or downwards will also determine the two-dimensional position of the focus relative to the vertex.

Directrix of Parabola

The directrix of a parabola is a straight line that provides a little bit of balance or perspective to understand the shape better. Each point on the parabola is equidistant from the focus and this directrix line, ensuring the parabola maintains its perfect symmetric shape.

• In the general parabola equation \(y = 4ax^2\), the directrix is described by the line \(y = -a\).
• In our equation where \(a = 1\), the directrix is the line \(y = -1\).

It acts almost like a mirror line underneath the parabola from which distance measurements are taken. The distance from any point on the parabola to the directrix is equal to its distance to the focus. This relationship ensures that the unique curved geometry of the parabola is consistent and precise.

Axis of Symmetry

The axis of symmetry is an essential line that runs through the vertex of the parabola, dividing it into two mirror-image halves. It is crucial for understanding the symmetrical nature of the parabola. In simpler words, it’s like the backbone of the parabola around which the shape is drawn.

• For a parabola defined by \(y = 4ax^2\), regardless of the value of "a", the axis of symmetry is always the line \(x = 0\).

The equation \(x = 0\) suggests that the parabola is super symmetric about the vertical y-axis. The axis of symmetry passes through the vertex of the parabola, which is at the origin \((0,0)\) in this case. This line is crucial as it ensures that for every point on one side of the parabola, there is an identical point on the opposite side equidistant from this axis.