Problem 22 - Calculus Single Variable

Question

The Cartesian equation of a parabola is given. Determine its vertex and axis of symmetry. \(y=2(3-x)^{2}+4\)

Step-by-Step Solution

Answer

Vertex: (3, 4); Axis of symmetry: x = 3.

Step 1: Identify the Equation Form

The given equation is in the form \[ y = a(x - h)^2 + k \]where \( a = 2 \), \( h = 3 \), and \( k = 4 \). In this form, it is easy to identify the vertex and axis of symmetry.

Step 2: Determine the Vertex

From the standard form, the vertex \((h, k)\) of the parabola is directly available. Here, \( h = 3 \) and \( k = 4 \), so the vertex of the parabola is at the point \( (3, 4) \).

Step 3: Establish the Axis of Symmetry

The axis of symmetry for a parabola in the form \( y = a(x - h)^2 + k \) is a vertical line given by the equation \( x = h \). For this parabola, since \( h = 3 \), the axis of symmetry is the line \( x = 3 \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a significant point. It is the highest or lowest point on the graph, depending on the parabola's orientation. In mathematical terms, the vertex is where the parabola changes direction.

When a parabola is expressed in the standard form \(y = a(x - h)^2 + k\), the vertex can be easily identified as \((h, k)\). The values of \(h\) and \(k\) correspond to the coordinates of the vertex. In our exercise, with the equation \(y = 2(3-x)^2 + 4\), the vertex is located at the point \((3, 4)\).

This implies that if you were to plot the parabola on a graph, the point \(3, 4\) would be where the curve peaks or dips. This particular point gives the parabola a visible and clear starting point in terms of understanding its shape. Always remember that the vertex is a central feature defining the structure of a quadratic graph.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex. This line helps maintain the symmetrical nature of the parabola.

For a parabola given in the standard form \(y = a(x - h)^2 + k\), the axis of symmetry can be described by the vertical line \(x = h\). Here, \(h\) is the \(x\)-coordinate of the vertex. In our specific example, \(h = 3\), so the axis of symmetry is \(x = 3\).

This concept is very useful because it not only helps in sketching the parabola accurately, but also in understanding its balance. Knowing the axis of symmetry allows you to determine the behavior of the parabola on either side of this line. It is invariably at the heart of the parabola, ensuring its perfect consistency and harmony.

Standard Form of a Parabola

The standard form of a parabola is particularly beneficial due to its simplicity and directness in expressing essential features like the vertex and axis of symmetry. It is given by the equation \(y = a(x - h)^2 + k\).

In this format:

• \(a\) indicates the parabola's opening direction and its width. A positive \(a\) value means the parabola opens upwards, and a negative value means it opens downwards.
• \(h\) is the \(x\)-coordinate of the vertex, which also determines the axis of symmetry.
• \(k\) is the \(y\)-coordinate of the vertex, setting the height at which the vertex is situated.

This form is particularly insightful because it allows one to determine the vertex and axis of symmetry immediately without further calculation, just as we solved for \(y = 2(3-x)^2 + 4\). Converting a quadratic equation to this form can make graphing and analyzing the parabola much simpler. It's like having a clear map to follow in comprehending the behavior of the quadratic graph.