Problem 86 - Precalculus with Limits

Question

CAPSTONE Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) \(y=a(x-h)^2+k, \quad a \neq 0\) (b) \((x-h)^2=4p(y-k), \quad p \neq 0\) (c) \((y-k)^2=4p(x-k), \quad p \neq 0\)

Step-by-Step Solution

Answer

Equation (a) represents a parabola in vertex form with its vertex at \((h, k)\). Equation (b) also represents a parabola in standard form with its vertex at \((h, k)\). These two equations are equivalent when \(4p=a\), as this relates the two different parameters of the two forms. Equation (c) represents a circle or an ellipse centered at \((h, k)\).

Step 1 - Understand the vertex form of a parabola

The first equation is the vertex form for a parabola: \(y=a(x-h)^2+k\), where \(a \neq 0\). In this form, the vertex of the parabola is \((h,k)\) and \(a\) determines the direction and the width of the parabola. If \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards.

Step 2 - Understand the standard form of a parabola

The second equation is the standard form for a parabola: \((x-h)^2=4p(y-k)\), where \(p \neq 0\). In this form, the vertex of the parabola is again \((h,k)\), and the parameter \(p\) relates to the focus and the directrix of the parabola. If \(p\) is positive, the parabola opens upwards; if \(p\) is negative, it opens downwards. However, it's important to note that the relation \(4p=a\) must be satisfied for these two forms to be equivalent.

Step 3 - Understand the form of an ellipse or circle

The third equation is the form for a circle or an ellipse: \((y-k)^2=4p(x-h)\), where \(p \neq 0\). If \(p=1\), the equation is a circle centered at \((h,k)\) with radius equal to \(sqrt{4p}\).If \(p \neq 1\), it is an ellipse with its center at \((h,k)\).

Step 4 - Establish the equivalence

Equations (a) and (b) are equivalent if the parameter \(p\) in equation (b) is chosen such that \(4p=\frac{1}{a}\). Then, given a certain \(x\)-value, both equations will yield the same \(y\)-value, which makes them equivalent.

Key Concepts

Parabola

A parabola is a U-shaped curve that is a hallmark of quadratic equations. It has a distinct, symmetrical arc that opens either upwards or downwards, depending on the equation. One interesting characteristic of a parabola is its vertex, which is the maximum or minimum point of the curve. Parabolas appear in many real-world scenarios, such as the path of a thrown ball or the design of some bridges.

• The vertex of a parabola is its highest or lowest point.
• The direction of opening (upward or downward) is determined by the equation's coefficients.
• Parabolas are symmetrically balanced around their vertex.

Understanding parabolas is essential in algebra and helps in analyzing graphs of quadratic functions.

Vertex Form

The vertex form of a parabola is given as \(y=a(x-h)^2+k\). This form is particularly useful because it directly reveals the vertex of the parabola, \((h, k)\). In this equation:

• \(h\) and \(k\) are the coordinates of the vertex.
• \(a\) affects the width and direction of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• The value of \(a\) also determines how "wide" or "narrow" the parabola appears.

By focusing on the vertex form, you can easily graph a parabola and identify key features like the axis of symmetry, which is the vertical line \(x = h\).

Standard Form

The standard form for a parabola centered at \((h, k)\) is \((x-h)^2=4p(y-k)\). This equation is another way to represent a parabola, emphasizing its geometric properties like the focus and directrix. Here's what each component represents:

• The vertex is still \((h,k)\).
• \(p\) measures the distance from the vertex to the focus and directrix.
• Positive \(p\) indicates an upward opening, while negative \(p\) suggests a downward opening.

The standard form is useful for solving problems involving optics or satellite dishes, where the distance from the vertex to other parts of the parabola is crucial.

Ellipse

An ellipse is an elongated circle, appearing as a flattened oval. The equation \((y-k)^2=4p(x-h)\) depicts an ellipse — or a circle if \(p=1\). This shape is common in astronomy and physics, particularly describing planetary orbits.

• When \(p = 1\), the equation represents a circle.
• If \(p eq 1\), it describes an ellipse.
• The center of the ellipse or circle is \((h,k)\).

Ellipses have unique properties like two focal points and major and minor axes, which affect how the shape stretches. Understanding ellipses is important in higher-level math and physics for studying or analyzing orbits and wave patterns.