Problem 65 - A Graphical Approach to College Algebra

Question

For the graph of \((x-h)^{2}=4 c(y-k)\), in what quadrant is the vertex for each condition? (a) \(h<0, k<0\) (b) \(h<0, k>0\) (c) \(h>0, k<0\) (d) \(h>0, k>0\)

Step-by-Step Solution

Answer

(a) Quadrant III, (b) Quadrant II, (c) Quadrant IV, (d) Quadrant I.

Step 1: Understanding the Equation

The equation \((x-h)^2 = 4c(y-k)\) describes a parabola. In this form, \( (h, k) \) represents the vertex of this parabola.

Step 2: Identifying the Quadrants

Recall that in the Cartesian plane, the four quadrants are defined as follows: 1. Quadrant I: Both \(x\) and \(y\) are positive. 2. Quadrant II: \(x\) is negative, \(y\) is positive. 3. Quadrant III: Both \(x\) and \(y\) are negative. 4. Quadrant IV: \(x\) is positive, \(y\) is negative.

Step 3: Condition (a): h<0, k<0

Given \( h<0 \) and \( k<0 \), the vertex \((h, k)\) has both a negative \(x\)-coordinate and a negative \(y\)-coordinate, which places it in Quadrant III.

Step 4: Condition (b): h 0

For \( h<0 \) and \( k>0 \), the vertex \((h, k)\) has a negative \(x\)-coordinate and a positive \(y\)-coordinate, situating it in Quadrant II.

Step 5: Condition (c): h>0, k<0

Here, \( h>0 \) and \( k<0 \) make the vertex \((h, k)\) have a positive \(x\)-coordinate and a negative \(y\)-coordinate, placing it in Quadrant IV.

Step 6: Condition (d): h>0, k>0

With \( h>0 \) and \( k>0 \), both coordinates of the vertex \((h, k)\) are positive, locating it in Quadrant I.

Key Concepts

Vertex of a Parabola

The vertex is a crucial point on a parabola that determines its position in the Cartesian plane. In the equation \((x-h)^2 = 4c(y-k)\), the vertex is given by the coordinates \((h, k)\). It represents the point where the parabola changes direction. Whether the curve opens upwards or downwards depends on the value of \(c\), but the vertex is always the pivot point for the parabola's shape.

To fully understand the vertex's role, consider some key points:

• The vertex is the minimum or maximum point of the parabola, depending on its orientation.
• It helps in graphing the parabola accurately by providing a reference point.
• The coordinates \((h, k)\) determine where the vertex is located relative to the origin of the Cartesian plane.

The equation form \((x-h)^2 = 4c(y-k)\) specifies that moving the vertex depends on \(h\) and \(k\). This means shifting the graph horizontally and vertically based on these values.

Understanding Quadrants

In mathematics, quadrants are crucial for pinpointing locations on a Cartesian plane. There are four quadrants, and knowing them can help you determine where a point like a vertex lies.

• Quadrant I has both coordinates positive, meaning any point here has \(x > 0\) and \(y > 0\).
• Quadrant II has a negative \(x\) and a positive \(y\), which is where any \((x, y)\) with \(x < 0\) and \(y > 0\) is located.
• Quadrant III is home to negative coordinates on both axes, so \((x, y)\) points here have \(x < 0\) and \(y < 0\).
• Quadrant IV, conversely, has a positive \(x\) and negative \(y\), meaning any \((x, y)\) will be \(x > 0\) and \(y < 0\).

This knowledge helps when analyzing a parabola and its vertex's position on the plane, relating directly to the conditions provided in problems like determining which quadrant the vertex falls into.

Introduction to the Cartesian Plane

A Cartesian plane is a two-dimensional coordinate system that facilitates plotting mathematical functions and points in a defined space. It consists of horizontal (\(x\)-axis) and vertical (\(y\)-axis) lines that intersect at the origin \((0,0)\). This creates a plane divided into four quadrants which are useful for graphical analysis.

The Cartesian plane serves many purposes:

• It provides a clear visual representation of algebraic expressions and geometric figures.
• It helps in graphing parabolas, making it easier to analyze their properties such as direction and vertex location.
• By understanding its axes and quadrants, one can determine the position of any point, including where the parabola's vertex falls, helping to solve complex mathematical problems with ease.

Overall, its structured setup with coordinates allows intricate mathematical relationships to be simplified by plotting them visually, making the Cartesian plane an essential tool in mathematics.