Problem 15 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ (y-4)^{2}=9(x-4) $$

Step-by-Step Solution

Answer

The equation is a parabola in standard form \((y-4)^2 = 9(x-4)\)\, opening to the right with vertex at (4,4).

Step 1: Identify the Type of Conic Section

To identify the conic section, let's compare the equation to the standard form of various conic sections. The given equation is \[(y-4)^2 = 9(x-4)\]This equation is of the form \[(y-k)^2 = 4p(x-h)\]which represents a parabola. Since it matches the format for a parabola, the graph is of a parabola.

Step 2: Write in Standard Form of Parabola

The standard form of a parabola that opens horizontally is \[(y-k)^2 = 4p(x-h)\]For the given equation \[(y-4)^2 = 9(x-4)\]we can see that it's already in the standard form with \[4p = 9\] leading to \[p = \frac{9}{4}\].This confirms it is a parabola with vertex at \((4, 4)\) and opens to the right since \(p\) is positive.

Step 3: Graph the Equation

Now, to graph the parabola, identify the vertex and focus. The vertex is at \((4, 4)\). Since it is a horizontal parabola that opens to the right, calculate the focus by moving \(\frac{9}{4}\) units to the right of the vertex.Thus, the focus is at \((4 + \frac{9}{4}, 4)\) or \((\frac{25}{4}, 4)\).Draw the axis of symmetry, which is a horizontal line through \(y = 4\), and sketch the parabola opening to the right.

Key Concepts

Standard Form

The standard form of a conic section is crucial for identifying and graphing curves like parabolas, circles, ellipses, and hyperbolas. In mathematical terms, the standard form is a way of arranging an equation so it highlights important features of the curve.
\[(y-k)^2 = 4p(x-h) \]This is the standard form for a parabola that opens horizontally. Here:

• \((h, k)\) is the vertex of the parabola.
• \(p\) is a parameter that determines the distance from the vertex to the focus.

For our given equation
\[(y-4)^2 = 9(x-4)\]we see it's already in standard form. This means minimal work is required to interpret and graph it.

Parabola

A parabola is a specific type of curve, part of the conic sections, and is represented in its standard form by:
\[(y-k)^2 = 4p(x-h) \]As derived, our equation fits into this form, identifying it as a parabola. Parabolas have unique properties:

• They are symmetrical, with the axis of symmetry as a pivotal line.
• They open either to the right, left, up, or down, depending on the equation.

Our parabola, \[(y-4)^2 = 9(x-4)\]opens horizontally to the right because \(p = \frac{9}{4}\) is positive. Understanding these defining aspects of parabolas helps in graphing them accurately.

Vertex

The vertex is a critical point on a parabola. It serves as the "turning point," from which the curve extends symmetrically. The vertex in the standard form equation:
\[(y-k)^2 = 4p(x-h)\]is located at point \((h, k)\).In our specific equation \[(y-4)^2 = 9(x-4)\],the vertex is:

• \((h, k) = (4, 4)\)

This means the parabola is centered around this point on the graph. Locating the vertex is foundational for graphing the parabola correctly and understanding its spatial orientation.

Focus

The focus of a parabola is a point inside the curve where distances from points on the parabola to the focus are equal when measured perpendicularly to the axis of symmetry. It plays a major role:

• In determining the direction and width of the parabola).
• In reflecting the parabolic trajectory's light or signals.

To find the focus for our parabola \[(y-4)^2 = 9(x-4)\],we use the vertex and the parameter \(p = \frac{9}{4}\). The focus is located \(p\) units away from the vertex in the direction the parabola opens.

• Since this parabola opens to the right (horizontally), the focus is at \((4 + \frac{9}{4}, 4)\), or \((\frac{25}{4}, 4)\).

Identifying the focus aids in sketching the parabola with precision and enhances understanding of its geometric structure.