Problem 37 - Intermediate Algebra

Question

(a) Graph \(y=x^{2}, y=(x-2)^{2}, y=(x-3)^{2}\), and \(y=\) \((x-5)^{2}\) on the same set of axes. (b) Graph \(y=x^{2}, y=(x+1)^{2}, y=(x+3)^{2}\), and \(y=\) \((x+6)^{2}\) on the same set of axes.

Step-by-Step Solution

Answer

Graphs show parabolas with different horizontal shifts.

Step 1: Understand the Basic Graph

The graph of \( y = x^2 \) is a parabola with its vertex at the origin (0,0). It opens upwards, and its basic shape is symmetric about the y-axis.

Step 2: Translate Horizontally for Part (a)

For \( y = (x-2)^2 \), \( y = (x-3)^2 \), and \( y = (x-5)^2 \), you are shifting the basic parabola \( y = x^2 \) horizontally to the right by 2, 3, and 5 units respectively. The graphs keep the same shape.

Step 3: Graph Parabolas for Part (a)

Graph \( y = x^2 \), \( y = (x-2)^2 \), \( y = (x-3)^2 \), and \( y = (x-5)^2 \) on the same set of axes. Each graph is a parabola opening upwards, with vertices at (0,0), (2,0), (3,0), and (5,0) respectively.

Step 4: Translate Horizontally for Part (b)

For \( y = (x+1)^2 \), \( y = (x+3)^2 \), and \( y = (x+6)^2 \), you are shifting the basic parabola \( y = x^2 \) horizontally to the left by 1, 3, and 6 units respectively.

Step 5: Graph Parabolas for Part (b)

Graph \( y = x^2 \), \( y = (x+1)^2 \), \( y = (x+3)^2 \), and \( y = (x+6)^2 \) on the same set of axes. Each graph is a parabola opening upwards, with vertices at (0,0), (-1,0), (-3,0), and (-6,0) respectively.

Key Concepts

Vertex Form

When graphing parabolas, one of the most insightful tools is the vertex form. The vertex form of a quadratic function is given by \[y = a(x-h)^2 + k\]Here, \((h, k)\) denotes the vertex, the highest or lowest point depending on the parabola's orientation. This format clearly shows the vertex, making it easier to understand translations and transformations. In the original exercise, you encounter such equations, helping you to determine the points where the parabola changes direction. Remember, the vertex is not just a point, it is where the graph symmetrically shifts on either side. Knowing this can help you make sense of how the parabola is oriented depending on the given values of \(h\) and \(k\).

Horizontal Translation

Horizontal translation in graphing is akin to physically shifting the graph left or right on the coordinate plane. It is crucial for understanding how equations in the form of \[y = (x - h)^2\]move. In part (a) of the exercise, changing from \( y = x^2 \) to \( y = (x - 2)^2 \) means moving the graph to the right by 2 units. Conversely, in part (b), \( y = (x + 1)^2 \) translates the graph to the left by 1 unit.

• Rightward Translation: Occurs when you have a negative sign (e.g., \((x-2)^2\)).
• Leftward Translation: Occurs with a positive sign (e.g., \((x+1)^2\)).

The shape of the graph remains unchanged during these translations; only the position shifts horizontally. Understanding these subtleties ensures you're adept at predicting movements based on equation changes.

Graphing Techniques

Graphing techniques for parabolas revolve around accurate plotting and recognizing transformations. The exercise guides you through translating these key movements:

• Starting Point: Begin with the basic graph \( y = x^2 \). Its vertex is at the origin \((0,0)\), forming a symmetrical U-shape.
• Plot Vertices: For additional equations, identify vertices like \((2,0)\) for \( y = (x-2)^2 \). Mark these vertices on the graph first.
• Connect Points: Draw smooth curves through these points, ensuring that the parabolas open upwards and maintain symmetry.

Each parabola maintains the same shape; only their positions change. Practicing such techniques can solidify your understanding, enabling your confidence in handling more complex quadratic transformations. Consistency in plotting ensures precision and gives you a visual feel of algebraic manipulations.