Problem 14 - 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. $$ y+4=(x-2)^{2} $$

Step-by-Step Solution

Answer

The given equation represents a parabola.

Step 1: Identify the Current Form

The equation given is \( y + 4 = (x - 2)^2 \). This is in the form \( y = (x - h)^2 + k \), which is a standard form for a parabola.

Step 2: Convert to Standard Form

We need to express the equation in the standard form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Start by simplifying and rearranging: \[ y + 4 = x^2 - 4x + 4 \] Subtract 4 from both sides to isolate zero on one side: \[ y = x^2 - 4x \] Thus, the standard form is: \[ x^2 - 4x - y = 0 \]

Step 3: Classify the Conic Section

Since there is only \( x^2 \) term without any \( y^2 \) term, and there is no \( Bxy \) term, this equation represents a parabola.

Step 4: Graph the Equation

To graph the parabola represented by the equation \( y = (x - 2)^2 - 4 \): - Identify the vertex: \( (h, k) = (2, -4) \) from \( y = (x - 2)^2 - 4 \). - Sketch the parabola opening upwards with vertex at (2, -4). - Plot additional points by selecting \( x \) values around the vertex (e.g., \( x = 1, 3 \)) and calculating corresponding \( y \) values.

Key Concepts

Understanding the Parabola

A parabola is a unique U-shaped curve that appears in various mathematical contexts, particularly in algebra and calculus. It's especially prevalent in quadratic functions. These curves are defined by equations of the form \( y = ax^2 + bx + c \), where the key feature is that the variable \( x \) is squared. This squaring gives the parabola its distinct shape.

Parabolas can open upward or downward, depending on the coefficient of the \( x^2 \) term. If the coefficient is positive, the parabola opens upwards, resembling a cup. If negative, it opens downwards, like an umbrella.

• Vertex: The lowest or highest point on the parabola, depending on its opening direction.
• Axis of Symmetry: A vertical line that runs through the vertex, dividing the parabola into mirror images.

Understanding these properties helps in graphing and analyzing parabolas in various mathematical problems.

Exploring Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. Each shape emerges from how the plane cuts through the cone. Parabolas form when the plane is parallel to the cone's edge.

In terms of equations:

• Circle: Both \( x^2 \) and \( y^2 \) appear with the same coefficient.
• Ellipse: Both \( x^2 \) and \( y^2 \) appear with different coefficients but are the same sign.
• Parabola: Only one of \( x^2 \) or \( y^2 \) appears in the equation.
• Hyperbola: Both \( x^2 \) and \( y^2 \) have coefficients of opposite signs.

Understanding these characteristics allows us to identify and classify the different conics in standard form quickly.

Graphing Equations Made Simple

Graphing equations involves plotting points on a coordinate plane to visualize mathematical relationships. For parabolas, this process begins by identifying the vertex, which is the turning point of the curve. For instance, in the equation \( y = (x - 2)^2 - 4 \), the vertex is \((2, -4)\).

After locating the vertex, you can sketch the parabola by selecting nearby \( x \)-values and calculating their corresponding \( y \)-values to find additional points.

• Select points symmetrically around the vertex for accuracy.
• Draw a smooth curve that goes through the points.

This method ensures a precise representation of the quadratic function, aiding analysis and interpretation in real-world problems or further mathematical exploration.

Unraveling Vertex Form

The vertex form of a quadratic equation is expressed as \( y = a(x - h)^2 + k \). This form is particularly useful because it makes the equation's vertex easily recognizable, \((h, k)\), shedding light on the graph's positioning and shape.

The "\( a \)" in the equation influences the parabola's width and direction.

• If \( a \) is positive, the parabola opens up; if negative, it opens down.
• A larger \( a \) makes the parabola narrower, while a smaller \( a \) makes it wider.

Knowing how to convert a standard quadratic equation into the vertex form can greatly simplify the process of graphing or analyzing the parabola. The vertex form comes in handy for quickly finding the minimum or maximum values of quadratic functions.