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The slope-intercept form is a way to write the equation of a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept. In our given function, \( h(x) = \frac{1}{2} x + 2\), the slope \( m \) is \( \frac{1}{2} \) and the y-intercept \( b \) is 2.

The slope tells us how steep the line is. For instance, a slope of \( \frac{1}{2} \) means that for every 2 units you move to the right (along the x-axis), you move 1 unit up (along the y-axis). This 'rise over run' approach helps in plotting the graph easily.

The y-intercept is the point where the line crosses the y-axis. In this case, it is at \( (0, 2) \). Knowing both the slope and y-intercept allows us to plot the linear function accurately on a graph.

When discussing the domain and range of a function, we are looking at the inputs and outputs of the function respectively.

The **domain** of a function includes all the possible x-values that you can input into the function. For linear functions like \( h(x) = \frac{1}{2} x + 2 \), the domain is all real numbers \( (-\infty, \infty) \). This means you can plug in any real number for x, and the function will still work.

The **range** of a function includes all the possible y-values that the function can output. For the function \( h(x) = \frac{1}{2} x + 2 \), the range is also all real numbers \( (-\infty, \infty) \). In other words, as x takes on any real number value from negative to positive infinity, y will also take on any real number value.

Plotting points is a fundamental skill in graphing linear functions. Let's break down the process using our function \( h(x) = \frac{1}{2} x + 2 \).

First, identify the y-intercept, which is the point where the line crosses the y-axis. From our function, this is \( (0, 2) \). Plot this point on the graph.

Next, use the slope to find another point. The slope \( \frac{1}{2} \) tells you to rise 1 unit for every 2 units you run to the right. Starting at \( (0, 2) \), move up 1 unit and right 2 units to locate the point \( (2, 3) \). Plot this point as well.

Finally, draw a straight line through the points \( (0, 2) \) and \( (2, 3) \). This is the graph of the linear function \( h(x) = \frac{1}{2} x + 2 \). By following these steps, you will be able to accurately represent the function on a graph.