Rigid Transformations: Analyzing $y+5=-2(x-1)^2$

Let's break down the equation $y+5 = -2(x-1)^2$ and figure out the transformations applied to the basic quadratic function. This involves understanding how the equation relates to shifts, reflections, and stretches of the original graph. We will look at each component individually to determine the nature of the rigid transformations, focusing on shifts and reflections, and then discuss the non-rigid transformation related to stretching. This will give you a comprehensive grasp of how to interpret quadratic equations and their corresponding graphical transformations.

Deconstructing the Equation

To accurately identify the transformations, we need to rewrite the equation in a more standard form: $y = -2(x-1)^2 - 5$. This form makes it easier to see each transformation applied to the basic quadratic function, $y = x^2$. The transformations here involve horizontal and vertical shifts, reflection across the x-axis, and a vertical stretch. Each of these affects the position and shape of the parabola. Understanding these transformations is vital for quickly sketching graphs and solving related problems. For instance, knowing how the vertex shifts can help in determining maximum or minimum values of the function, a common application in optimization problems.

Horizontal Shift

The term $(x-1)$ inside the parenthesis indicates a horizontal shift. Specifically, it means the graph is shifted 1 unit to the right. Remember, it's the opposite of what you might initially think. If it were $(x+1)$, the shift would be 1 unit to the left. This is a fundamental concept in understanding function transformations. Horizontal shifts directly affect the x-coordinate of every point on the graph, moving the entire parabola to the right or left. The vertex of the basic parabola at (0,0) is now at (1,0) before considering other transformations. This understanding is crucial when dealing with transformations of any function, not just quadratics.

Vertical Shift

The term $-5$ outside the parenthesis indicates a vertical shift. This means the entire graph is shifted 5 units downward. This transformation is more intuitive; a negative number shifts the graph down, and a positive number shifts it up. Vertical shifts affect the y-coordinate of every point on the graph, moving the entire parabola up or down. After this shift, the vertex, which was at (1,0), is now at (1,-5). Understanding vertical shifts is essential in various applications, such as determining the range of a function or finding the minimum/maximum value in applied problems.

Reflection and Vertical Stretch

The coefficient $-2$ in front of the parenthesis combines two transformations. The negative sign indicates a reflection across the x-axis. This flips the parabola upside down. The '2' indicates a vertical stretch by a factor of 2. This makes the parabola skinnier compared to the basic parabola $y = x^2$. Reflections change the sign of the y-coordinate of each point, while vertical stretches multiply the y-coordinate by the stretch factor. This combined transformation significantly alters the shape and orientation of the parabola, impacting its overall appearance and characteristics. The vertex remains at (1, -5), but the parabola now opens downwards due to the reflection.

Analyzing the Options

Now, let's look at the original question's options in light of our analysis:

A. The graph is shifted 1 unit left. B. The graph is reflected across the $y$-axis. C. The graph is stretched vertically by a factor of -2.

Based on our deconstruction of the equation, option A is incorrect because the graph is shifted 1 unit to the right, not left. Option B is also incorrect; the graph is reflected across the x-axis, not the y-axis. Option C is partially correct; the graph is stretched vertically by a factor of 2, and the negative indicates a reflection about the x-axis, but it doesn't fully describe the whole transformation. The original question only asks about the rigid transformation.

Identifying Rigid Transformations

Rigid transformations are those that do not change the shape or size of the figure. These include translations (shifts) and reflections. Stretches and compressions are not rigid transformations because they alter the shape. In our equation, $y+5 = -2(x-1)^2$, the rigid transformations are the horizontal shift of 1 unit to the right, the vertical shift of 5 units down, and the reflection across the x-axis. The vertical stretch by a factor of 2 is a non-rigid transformation.

Considering only the rigid transformations, the correct answer would be the shift of 1 unit to the right and the shift of 5 units down. While the reflection is also a rigid transformation, it wasn't explicitly given as an option.

Conclusion

Understanding transformations of functions, particularly quadratic functions, is a fundamental concept in algebra and calculus. By breaking down the equation $y+5 = -2(x-1)^2$, we identified a horizontal shift of 1 unit to the right, a vertical shift of 5 units down, a reflection across the x-axis, and a vertical stretch by a factor of 2. Recognizing which transformations are rigid (shifts and reflections) and which are not (stretches and compressions) is essential for accurately describing how the graph of the function is altered. This knowledge enables you to quickly sketch graphs, solve related problems, and deepen your understanding of function behavior. Remember to pay close attention to the signs and coefficients in the equation, as they dictate the direction and magnitude of each transformation.