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Linear functions are the simplest type of functions that create a straight line when graphed. The general form for a linear function is given by the equation \( f(x) = mx + b \), where:

• \( m \) is the slope of the line, representing the rate of change.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For example, the function \( f(x) = x \) is linear with a slope of 1 and a y-intercept of 0. This means that for every 1 unit increase in \( x \), \( f(x) \) increases by 1 unit. Linear functions are foundational in understanding the concepts of function transformations, such as shifts and compressions.

Vertical compression is a transformation that "squashes" the graph of a function towards the x-axis. It is achieved by multiplying the entire function by a factor between 0 and 1.Suppose you have a function \( f(x) = x \) and apply a vertical compression by \( \frac{1}{3} \). This results in the new function \( h(x) = \frac{1}{3}x \). Here, each output of the function is one third of what it originally was, making the graph look flatter.Whenever you see a vertical compression, think of it as decreasing the steepness of the line or curve while keeping the x-values unaffected. The slope, as a result, reduces by the compression factor, highlighting the function's rate of change less dramatically.

A horizontal shift moves the graph of a function left or right along the x-axis. It is achieved by adding or subtracting a constant to the input variable \( x \).In our example, after compressing the function, we apply a horizontal shift by shifting it left by 1 unit. To do this, replace \( x \) by \( x + 1 \) in the function \( h(x) = \frac{1}{3}x \). This transformation leads to the equation \( h(x) = \frac{1}{3}(x + 1) \), effectively moving each point on the graph 1 unit to the left.Whether shifting left or right, remember:

• Add to \( x \) for a left shift.
• Subtract from \( x \) for a right shift.

Using this method, you can control where the graph starts along the x-axis, allowing for precise position adjustments.

Vertical shifts adjust the graph of a function up or down along the y-axis. This process involves adding or subtracting a constant to the entire function.In our case, we are looking at moving the function up by 3 units after applying the horizontal shift and compression. Starting from the function \( h(x) = \frac{1}{3}(x + 1) \), add 3 to each output value: \( g(x) = \frac{1}{3}(x + 1) + 3 \).Simplifying the equation, we find \( g(x) = \frac{1}{3}x + \frac{10}{3} \).This final step involves adjusting all points upward, preserving the shape of the graph while changing the y-intercept. A vertical shift is one of the simplest transformations and is often used to align graphs with desired base levels.