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The slope-intercept form of a linear equation is a fundamental concept in graphing functions. It is given by the formula:\[ y = mx + b \]where:

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

This form makes it easy to quickly identify these two crucial aspects of a line. In our example, the function \( h(c) = -\frac{5}{2}c+4 \) is already in slope-intercept form. Here, the slope \( m \) is \(-\frac{5}{2}\), indicating how steeply the line rises or falls.
The y-intercept \( b \), on the other hand, is \(4\). This tells us that the line will pass through the y-axis at the point \((0, 4)\). By understanding this form, you can quickly sketch the line's basic characteristics, making subsequent steps like plotting points easier to manage.

Plotting points is a key step in graphing any function. It involves choosing values for the independent variable and calculating their corresponding dependent values.
To graph the function \( h(c) = -\frac{5}{2}c + 4 \), start by identifying the y-intercept \((0, 4)\), which is a given point on the line. After establishing this, choose a convenient value for the variable \( c \) to find additional points.
For example, select \( c = 2 \). Substitute it into the function to calculate:\[h(2) = -\frac{5}{2}(2) + 4 = -5 + 4 = -1\]Thus, the coordinate \((2, -1)\) is another point on the graph. Plotting several points can help ensure accuracy, especially if a mistake occurs. Each point reinforces the line's direction and slope.

A coordinate plane is a two-dimensional space formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, \((0,0)\). When graphing a linear function, such as \( h(c) = -\frac{5}{2}c + 4 \), it's important to properly plot each point on this plane.
The x-values represent the independent variable (in this case, \( c \)), and the y-values represent the dependent results from the function. Once you plot points like \((0, 4)\) and \((2, -1)\) on the coordinate plane, draw a straight line through them. This line illustrates all points satisfying the equation.
The coordinate plane helps visualize the relationship between the variables, facilitating a broader understanding of the function itself. It enables students to see how changes in \( c \) affect \( h(c) \), demonstrating both the slope and the intercept in a tangible way.