Question

CRITICAL THINKING Use the correct value \(0,-2\), or 1 with \(a, b\), and \(c\) so the graph of \(g(x)=a|x-b|+c\) is a reflection in the \(x\)-axis followed by a translation one unit to the left and one unit up of the graph of \(f(x)=2|x-2|+1\). Explain your reasoning.

Short answer

The values for \(a, b\), and \(c\) that make the graph of \(g(x)=a|x-b|+c\) a reflection in the x-axis followed by a translation one unit to the left and one unit up of the graph of \(f(x)=2|x-2|+1\) are \(a=-1\), \(b=1\), \(c=0\).

Step-by-step solution

Reflect the function

To reflect the graph of the function \(f(x)=2|x-2|+1\) in the x-axis, we need to negate the whole function. The reason is that reflecting in the x-axis changes the sign of the y-values of the function. Negating the whole function changes the function to \(f(x)=-2|x-2|-1\). That is, we replace the a in the equation \(g(x)=a|x-b|+c\) with -1.

Translate the function to the left

To translate the function one unit left, we need to replace \(x\) in the absolute value with \(x+1\), as replacing \(x\) with \(x-h\) shifts the graph h units to the right. In this case, our h is -1, because we are shifting the graph to the left. So, the function becomes \(f(x)=-2|x+1-2|-1\). That is, we replace the b in the equation \(g(x)=a|x-b|+c\) with 1.

Translate the function upwards

Finally, we need to translate the function one unit upwards. To achieve this, we add 1 to the entire function as addition affects vertical shift. This gives us \(g(x)=-2|x-2|+1+1\), which simplifies to \(g(x)=-2|x-1|+0\). That is, we replace the c in the equation \(g(x)=a|x-b|+c\) with 0.

Key concepts

Reflection in the x-axis

Understanding the concept of reflection in the x-axis is fundamental in algebra and function transformations. In simple terms, reflecting a function in the x-axis means flipping it over said axis, causing the positive y-values to become negative, and vice versa. Mathematically, we achieve this by multiplying the entire function by -1. For an absolute value function like \( f(x) = |x| \), a reflection would transform it into \( f(x) = -|x| \). This method applies to any function, and the same principle is utilized when working on more complex absolute value functions like in the given problem.

Horizontal translations

Horizontal translations are shifts of the graph of a function along the x-axis. In Algebra 2, such translations are critical for manipulating the position of a graph without altering its shape. When you replace \( x \) with \( x - h \) in your function's equation, you're telling your graph to move h units to the right. Conversely, if h is negative, as in \( x + h \) where h is a negative number, the graph moves h units to the left. This technique was employed in the example problem where the graph moved one unit to the left with \( x + 1 \).

Vertical translations

Much like their horizontal counterparts, vertical translations slide the graph of a function up or down along the y-axis without altering its shape. They're achieved by simply adding or subtracting a number from the entire function. When the number is positive, the graph ascends; when it's negative, the graph descends. For instance, \( f(x) = |x| + 3 \) would shift the graph of \( f(x) = |x| \) three units upwards. In our textbook problem, adding 1 resulted in the graph moving a single unit up.

Piecewise functions

Algebra 2 often introduces students to piecewise functions, which are defined by different expressions depending on the domain of the input value, x. These functions can exhibit completely different behaviors or follow distinct rules for different intervals of x. They are incredibly versatile and can model a wide range of real-world scenarios. When graphing, you would treat each piece of the function within its specified interval and combine these pieces to get the overall graph. Although the original exercise does not directly deal with piecewise functions, understanding them can help you grasp more complex transformations and combinations of functions.

Algebra 2

Algebra 2 students delve into the realm of function transformations, which includes reflections, translations, and more. The course lays the groundwork for mathematical thinking applied in calculus and beyond. The significance of mastering transformations lies in the ability to visualize and manipulate graphs to meet certain criteria, just as in the provided exercise where the combination of reflection and translations was used to modify the graph of an absolute value function.

When approaching problems of this nature, it's important to understand the order of operations for transformations. This ensures that each step is executed correctly, leading to the desired final graph. Absolute value transformations, in particular, are a staple in Algebra 2, providing a straightforward yet powerful example of how these principles are applied.