Problem 21 - Calculus Single Variable

Question

The Cartesian equation of a parabola is given. Determine its vertex and axis of symmetry. \(y=x^{2}-3\)

Step-by-Step Solution

Answer

Vertex: (0, -3), Axis of symmetry: x = 0.

Step 1: Identify the standard form

The given parabola equation is already in the standard form of a parabola, which is \( y = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = -3 \).

Step 2: Determine the vertex

For a parabola in the form \( y = ax^2 + bx + c \), the vertex can be calculated using the formula \( x = -\frac{b}{2a} \). Since \( b = 0 \) and \( a = 1 \), substitute these values into the formula: \( x = -\frac{0}{2 \times 1} = 0 \). Substituting \( x = 0 \) back into the equation \( y = x^2 - 3 \), we get \( y = 0^2 - 3 = -3 \). Therefore, the vertex is \((0, -3)\).

Step 3: Find the axis of symmetry

The axis of symmetry of a parabola \( y = ax^2 + bx + c \) is a vertical line that passes through the vertex. Its equation is \( x = -\frac{b}{2a} \). Using the values \( b = 0 \) and \( a = 1 \), as calculated previously, the axis of symmetry is \( x = 0 \).

Key Concepts

Vertex

The vertex of a parabola is a key point that represents either the minimum or maximum of the quadratic function, depending on whether the parabola opens upwards or downwards. The vertex is located at the very bottom (minimum) of a parabola that opens upwards, or at the very top (maximum) of one that opens downwards.
The formula to find the x-coordinate of the vertex is given by:

• \( x = -\frac{b}{2a} \)

This formula is derived from the standard form of a quadratic equation \( y = ax^2 + bx + c \). Once you have the x-coordinate, you substitute it back into the equation to find the y-coordinate of the vertex:

• \( y = a(x^2) + bx + c \)

In our case with the parabola equation \( y = x^2 - 3 \):

• We have \( a = 1 \), \( b = 0 \), so \( x = -\frac{0}{2 \times 1} = 0 \).
• The y-coordinate is determined by substituting x back: \( y = 0^2 - 3 = -3 \).

The vertex of the parabola is at the point \((0, -3)\). This suggests that the parabola opens upwards as the coefficient \( a \) is positive.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. This line helps us understand the symmetrical nature of parabolas.
The formula to find the axis of symmetry is the same as the x-coordinate of the vertex:

• \( x = -\frac{b}{2a} \)

In the context of the parabola \( y = x^2 - 3 \), since \( a = 1 \) and \( b = 0 \), the calculation for the axis of symmetry is straightforward as we already found in the vertex section:

• \( x = 0 \)

The equation of the axis of symmetry is therefore a vertical line at \( x = 0 \). This means that if you fold the parabola along this line, both halves would match perfectly. Understanding this helps in graphing parabolas accurately.

Standard form of a quadratic equation

The standard form of a quadratic equation is one of the most common ways to express a quadratic function. This form is expressed as:

• \( y = ax^2 + bx + c \)

Where:

• \( a \), \( b \), and \( c \) are constants.
• \( a \) determines the direction and width of the parabola.
• \( b \) and \( c \) influence the position of the parabola along the x and y axes respectively.

In our example with \( y = x^2 - 3 \), we have:\

• \( a = 1 \), \( b = 0 \), \( c = -3 \)

• This tells us the parabola opens upwards, is centered on the y-axis, and shifted downward by 3 units.

The form makes it easy to derive important features of the parabola, such as finding the vertex, axis of symmetry, and the direction in which the parabola opens. Recognizing and understanding this form is crucial in solving quadratic-related problems and graphing.