Problem 39 - Intermediate Algebra

Question

(a) Graph \(y=x^{2}-12 x+41\) and \(y=x^{2}+12 x+41\) on the same set of axes. What relationship seems to exist between the two graphs? (b) Graph \(y=x^{2}-8 x+22\) and \(y=-x^{2}+8 x-22\) on the same set of axes. What relationship seems to exist between the two graphs? (c) Graph \(y=x^{2}+10 x+29\) and \(y=-x^{2}+10 x-29\) on the same set of axes. What relationship seems to exist between the two graphs? (d) Summarize your findings for parts (a) through (c).

Step-by-Step Solution

Answer

The first pair shows vertical symmetry; the second and third pairs show symmetry around the horizontal axis.

Step 1: Analyze the equations from Part (a)

Identify the equations to be graphed: 1. \(y = x^2 - 12x + 41\) 2. \(y = x^2 + 12x + 41\).Both equations have similar structures with differences in the middle term, indicating they are symmetric about the y-axis translated by the constant term (41). You should notice that these two functions differ only in the coefficient of the linear term, suggesting reflection symmetry depending on changes in x-coefficients.

Step 2: Analyze the equations from Part (b)

Identify the equations to be graphed: 1. \(y = x^2 - 8x + 22\) 2. \(y = -x^2 + 8x - 22\).Notice that the second equation is a transformation of the first, with the leading coefficient of \(x^2\) being negative, indicating a reflection over the x-axis and inverts concavity. Both share the same coefficients for the linear term, suggesting some reflection or symmetry in a rotated sense.

Step 3: Analyze the equations from Part (c)

Identify the equations to be graphed: 1. \(y = x^2 + 10x + 29\) 2. \(y = -x^2 + 10x - 29\).Here, similar to Part (b), the negative coefficient for the quadratic term in the second equation indicates a reflection over the x-axis. The linear terms are equal so they reflect a similar symmetry relationship to Part (b).

Step 4: Graph the equations

Graph each pair of equations on their own coordinate system, making note of vertex positions and the parabolas' opening directions:- The first pair \((Part\ (a))\) are both upward-opening parabolas symmetrical around the y-axis influenced by quadratic and constant coefficients.- The second \((Part\ (b))\) and third pairs \((Part\ (c))\) involve upward and downward parabola symmetries, with corresponding turning points and rotations over the vertical axis because one equation in each pair has a negative leading term.

Step 5: Determine relationships from graphs

From your graphs: - Part (a): The parabolas are mirror images of each other when split by the y-axis. - Part (b): Each pair of parabolas reflects around the horizontal axis. - Part (c): Also reflects around the horizontal axis but displaced vertically and horizontally based on the linear and constant coefficients.

Step 6: Summarize Findings

Through graphing and comparing each pair: - In Part (a), these parabolas are reflections around vertical axes due to opposite linear coefficients. - In Parts (b) and (c), they reflect around the horizontal axis of the graph with reversed orientation concerning the opening direction. - Each pair maintains consistent transitional symmetry hints based on the leading and linear coefficients in each equation.

Key Concepts

Reflection Symmetry in Quadratic Functions

In mathematical terms, reflection symmetry for quadratic functions refers to how a graph can be flipped over a specific line and still resemble the original graph. This is particularly relevant when examining parabolas, which are the U-shaped curves often seen in quadratic functions. Reflection symmetry can arise in different ways depending on the coefficients in the quadratic equation.
For instance, consider the equations from part (a) of the original exercise:

• \(y = x^2 - 12x + 41\)
• \(y = x^2 + 12x + 41\)

You can observe that both equations have identical quadratic and constant terms but differ in the sign of the linear coefficient. This means that they are symmetric about the y-axis, where each is a mirror image of the other.
Similarly, in parts (b) and (c), reflection symmetry emerges slightly differently. Both involve one equation with a flipped sign in front of the quadratic term. This indicates a reflection over the x-axis, which flips the parabola vertically. Such transformations maintain the core integrity of the curve, preserving important features like the vertex and the axis of symmetry, albeit with a flipped orientation.

Understanding Parabolas

Parabolas are fundamental shapes in mathematics and appear frequently in quadratic functions. They are characterized by a distinct U-shape, with a vertex as the highest or lowest point, depending on whether the parabola opens upwards or downwards. The vertex also determines the axis of symmetry—a vertical line that divides the parabola into two mirror-image halves.
In the case of part (a) from the exercise:

• Both equations represent parabolas opening upward due to positive quadratic coefficients.
• The vertex can be found via completing the square or using vertex formulas.
• The parabolas from part (a) mirror each other across the y-axis, showing clear vertical symmetry.

In part (b) and (c), where a negative coefficient appears:

• We see one parabola open upwards and another open downwards.
• This occurs because of a negative sign before the quadratic term. It flips the parabola over the x-axis.

The concept of parabolas, with their axis of symmetry and vertex, provides a foundational understanding of how quadratic functions behave visually and algebraically.

Equation Analysis of Quadratic Functions

Analyzing the equations of quadratic functions in detail helps understand their graphical behaviour. Quadratics generally take the form \( y = ax^2 + bx + c \), where:

• \( a \) dictates the parabola's opening direction and width.
• \( b \) influences the horizontal location of the vertex.
• \( c \) represents the y-intercept, where the parabola crosses the y-axis.

In the exercises:
Part (a) equations have different linear coefficients \( b = -12 \) and \( b = 12 \). This change produces a parabolic reflection across the y-axis without altering the opening direction because \( a \eq 0 \). Part (b) involves one equation where \( a \) takes on a negative value, \( a = -1 \). This causes the parabola to flip over the x-axis, demonstrating the impact of the leading coefficient's sign in the equation.
The consistency of the quadratic coefficients across pairs indicates parallel symmetry themes in these graphs. It shows how each term in the equation contributes uniquely to the overall shape and position of the parabola. Understanding these effects of each coefficient aids in predicting the transformation and reflection behavior of quadratic functions graphically. This makes graphing quadratics more intuitive and analytical.