Problem 62 - Algebra 2

Question

REVIEW Which of the following sentences is true about the graphs of \(y=2(x-3)^{2}+1\) and \(y=2(x+3)^{2}+1 ?\) F Their vertices are maximums G The graphs have the same shape with different vertices. H The graphs have different shapes with different vertices. J One graph has a vertex that is a maximum while the other graph has a vertex that is a minimum.

Step-by-Step Solution

Answer

G The graphs have the same shape with different vertices.

Step 1: Identify the form of the equation

Both equations given, \( y = 2(x-3)^2 + 1 \) and \( y = 2(x+3)^2 + 1 \), are in vertex form, which can be expressed as \( y = a(x-h)^2 + k \). Here, \( (h, k) \) represents the vertex of each parabola.

Step 2: Determine the vertex of each graph

For the equation \( y = 2(x-3)^2 + 1 \), the vertex \( (h, k) \) is \((3, 1)\). For the equation \( y = 2(x+3)^2 + 1 \), the vertex is \((-3, 1)\).

Step 3: Analyze the coefficient \(a\)

In both equations, the coefficient \( a = 2 \) is positive, which indicates that both parabolas open upwards, meaning their vertices are minimums, not maximums.

Step 4: Compare the shapes and vertices

Since the coefficients of the quadratic term are the same (both 2), the parabolas have the same shape; they are merely translated horizontally on the graph (different x-values for the vertices). Hence, they have the same shape but different vertices.

Step 5: Conclusion

Based on the analysis, the graphs have the same shape (since the coefficient \( a \) is the same), but they have different vertices because the \( h \) values in their vertex forms are different.

Key Concepts

Understanding Parabolas

A parabola is a U-shaped curve that appears when you graph a quadratic function. Any equation you can write in the form \( y = ax^2 + bx + c \) will generate a parabola when graphed. Parabolas have several distinctive properties:

• They can open upwards or downward depending on the coefficient \( a \), which determines the direction of the parabola. If \( a \) is positive, the parabola opens upwards; if it's negative, it opens downwards.
• The highest or lowest point of a parabola is called the vertex. This is a key feature of the parabola because it represents the maximum or minimum value of the quadratic function.
• Parabolas are symmetric along a vertical line that passes through their vertex, known as the axis of symmetry.

When working with quadratic functions, understanding how these properties interact allows us to predict the shape and position of the parabola before even graphing it.

Decoding Quadratic Functions

A quadratic function is a type of polynomial function where the highest power of the variable is squared. It is typically expressed in the form \( y = ax^2 + bx + c \). However, it can also be presented in vertex form, which is often easier to analyze for graphing purposes. The vertex form is written as \( y = a(x-h)^2 + k \), where \((h, k)\) describes the vertex of the parabola.

Key characteristics of quadratic functions include:

• The coefficient \( a \) determines the parabola's direction and how "wide" or "narrow" it appears. A larger \( a \) value makes the parabola narrower, while a smaller \( a \) value makes it wider.
• The vertex \((h, k)\) provides the minimum or maximum point of the graph, depending on whether the parabola opens upward or downward.
• The function's y-intercept occurs where \( x = 0 \), giving the point \((0, c)\) when in standard form. In vertex form, this is calculated after finding \( h \) and \( k \).

By understanding these attributes, students can plot quadratic functions onto a graph and understand their properties more thoroughly.

Grasping Graph Translation

Graph translation involves shifting, stretching, or flipping the graphs of functions across the Cartesian plane without altering their shape. For parabolas in vertex form, translations are easier to identify because they directly relate to the values of \( h \) and \( k \).

Translations affect graphs in the following ways:

• Horizontal translation: Changing \( h \) moves the graph left or right. If \( h \) increases, the graph shifts right. If \( h \) decreases, it shifts left.
• Vertical translation: Changing \( k \) shifts the graph up or down. Increasing \( k \) moves the graph up, while decreasing it moves the graph down.
• If the graph is reflected over the x-axis, that's known as a reflection, which occurs when \( a \) is negative.

For the functions \( y = 2(x-3)^2 + 1 \) and \( y = 2(x+3)^2 + 1 \), the horizontal translation results in the vertices at \((3, 1)\) and \((-3, 1)\) respectively. The parabolas themselves do not change shape, only their position shifts.