How are transformations of a linear function and an absolute value function similar? Provide two examples/ reasons.
Transformations of a linear function and an absolute value function are similar in the following ways:
1. Both types of functions can be shifted horizontally and vertically: The graphs of linear functions and absolute value functions can be shifted horizontally and vertically by adding or subtracting values inside the function. For example, consider the linear function f(x) = 2x + 3. Shifting this function horizontally by 2 units would result in f(x-2) = 2(x-2) + 3, and shifting it vertically by 3 units would result in f(x) + 3 = 2x + 6. Similarly, for an absolute value function, like f(x) = |x|, shifting it horizontally by 2 units would result in f(x-2) = |x-2|, and shifting it vertically by 3 units would result in f(x) + 3 = |x| + 3.
2. Both types of functions can be stretched or compressed vertically: The graphs of linear functions and absolute value functions can be stretched vertically (made steeper) or compressed vertically (made flatter) by multiplying or dividing by a value outside the function. For example, consider the linear function f(x) = 2x + 3. Multiplying this function by 2 would result in 2f(x) = 4x + 6, which would make the graph steeper. Similarly, for an absolute value function, like f(x) = |x|, multiplying it by 2 would result in 2f(x) = 2|x|, which would also make the graph steeper.
To understand the similarities between transformations of a linear function and an absolute value function, let's first briefly understand what these functions are.
A linear function is a function that has a constant rate of change and always has a straight-line graph. It can be represented in the form y = mx + b, where m is the slope and b is the y-intercept.
An absolute value function is a function that measures the distance of a number from zero on the number line. It is represented by f(x) = |x|, where |x| represents the absolute value of x.
Now, let's dive into the similarities between the transformations of these two functions:
1. Shifts Along the x-axis:
Both linear functions and absolute value functions can undergo shifts along the x-axis. For linear functions, a shift along the x-axis is represented by changing the value of b in the equation y = mx + b. If b is positive, the graph is shifted to the left. If b is negative, the graph is shifted to the right. Similarly, for absolute value functions, a shift along the x-axis is represented by changing the value inside the absolute value bars. If that value is positive, the graph is shifted to the left, and if it is negative, the graph is shifted to the right.
Example 1:
Linear function: y = 2x + 3
If we move the graph of this function 2 units to the left, it becomes y = 2(x+2) + 3, which results in a graph shifted to the left.
Absolute value function: f(x) = |x|
If we move the graph of this function 3 units to the right, it becomes f(x) = |x-3|, which results in a graph shifted to the right.
2. Reflections Over the x-axis:
Both linear functions and absolute value functions can be reflected over the x-axis. For linear functions, a reflection over the x-axis is represented by changing the sign of the slope (m) in the equation y = mx + b. For absolute value functions, a reflection over the x-axis is represented by multiplying the function by -1.
Example 2:
Linear function: y = -2x + 5
If we reflect the graph of this function over the x-axis, it becomes y = 2x + 5, resulting in a graph reflected over the x-axis.
Absolute value function: f(x) = |x|
If we reflect the graph of this function over the x-axis, it becomes f(x) = -|x|, resulting in a graph reflected over the x-axis.
Overall, both linear functions and absolute value functions can be transformed by shifting along the x-axis and reflecting over the x-axis. These transformations alter the position and orientation of the graphs, but the underlying structure of the functions remains the same.
the same way transformations of any two functions are similar
a*f(b(x-h))+k
does dilation and translation to any function f(x)