Problem 65 - A Graphical Approach to Precalculus with Limits

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) Third quadrant; (b) Second quadrant; (c) Fourth quadrant; (d) First quadrant.

Step 1: Identify the Vertex

The equation \((x-h)^2 = 4c(y-k)\) is in the form of the equation of a parabola, specifically a vertical parabola. The vertex of any parabola in this form is given by the point \(h, k\). Thus, the vertex of the given parabola is at the point \(h, k\)."

Step 2: Determine the Quadrant for h < 0, k < 0

In this case, the vertex \(h, k\) is \(h < 0, k < 0\), which means the vertex lies in the third quadrant since both coordinates \(x\) (represented by \(h\)) and \(y\) (represented by \(k\)) are negative.

Step 3: Determine the Quadrant for h 0

Here, the vertex \(h, k\) is \(h < 0, k > 0\). This indicates that the vertex is in the second quadrant as the \(x\) coordinate is negative and the \(y\) coordinate is positive.

Step 4: Determine the Quadrant for h > 0, k < 0

For this condition, the vertex \(h, k\) is \(h > 0, k < 0\). This tells us that the vertex is in the fourth quadrant because the \(x\) coordinate is positive while the \(y\) coordinate is negative.

Step 5: Determine the Quadrant for h > 0, k > 0

In this scenario, the vertex \(h, k\) becomes \(h > 0, k > 0\), placing the vertex in the first quadrant since both the \(x\) (\(h\)) and \(y\) (\(k\)) coordinates are positive.

Key Concepts

Vertex

In the world of parabolas, the **vertex** plays a central role. Imagine it as the peak or the lowest point of the curve, forming a kind of anchor for the parabola. The formula \( (x-h)^2 = 4c(y-k) \) reveals a vertical parabola, characterized by its curve opening either up or down. The values of \( h \) and \( k \) define the coordinates of this crucial vertex point, written as \( (h, k) \).

• The vertex represents the "turning point" where the parabola changes direction.
• Finding the vertex helps to graph the parabola accurately.
• It provides a reference to determine the orientation and position of the parabola.

Understanding the vertex is essential for sketching and problem-solving in coordinate geometry.

Quadrants

The coordinate plane is divided into four **quadrants**, each with its distinct positive or negative identifiers for the \( x \) and \( y \) axes:

• **First Quadrant**: Both \( x \) and \( y \) coordinates are positive. It’s the upper-right section of the plane.
• **Second Quadrant**: \( x \) is negative, \( y \) is positive. It covers the upper-left area.
• **Third Quadrant**: Both \( x \) and \( y \) are negative, positioned in the lower-left.
• **Fourth Quadrant**: \( x \) is positive, \( y \) is negative, located in the lower-right.

By examining the signs of \( h \) and \( k \) from the vertex \( (h, k) \), you can easily determine which quadrant the vertex falls into:

• If \( h > 0, k > 0 \), it’s in the first quadrant.
• If \( h < 0, k > 0 \), second quadrant.
• If \( h < 0, k < 0 \), third quadrant.
• If \( h > 0, k < 0 \), fourth quadrant.

Understanding the relationship between a parabola's vertex and the quadrants helps in visualizing how different transformations shift the parabola across the plane.

Coordinate Geometry

In **coordinate geometry**, points, lines, and curves are defined using a coordinate plane, which significantly enhances your ability to visualize complex relationships. Parabolas, like the one defined by \( (x-h)^2 = 4c(y-k) \), are a classic example where coordinate geometry is applied.
This field connects algebra with geometry through coordinates on a plane:

• Coordinates are used to pinpoint the exact location of points within the plane.
• The plane is a combination of two axes, the horizontal \( x \)-axis and the vertical \( y \)-axis, which intersect at the origin \((0,0)\).
• Shapes and their properties, such as parabolas, are analyzed by examining equations and their graphical representations.

Coordinate geometry allows you to explore how the equation's parameters affect the graph's shape and position. For example, in our specific parabola, changing \( h \) and \( k \) shifts its vertex, illustrating dynamic interconnectedness between an equation's variables and its graphical output. Understanding these elements is crucial for mastering the concepts involved, making problem-solving more intuitive and less abstract.