Initial Value

Essential Question for Topic

How do you know a linear function when you see one?

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Focus Question for Lesson

Each part of a linear function plays a role in how its graph looks. What role does the initial value play?

Standards

8.F.A.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

8.F.B.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.B.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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We want to know where the function starts initially. What is the value of the function at the input of zero. We call this the initial value.

Graphically it is easy to see the initial value because it is where the graph crosses the y-axis.

The values represented in the function can be interpreted in the following way:

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y=mx +b; b is the y-intercept, initial value

The coefficient of 𝒙 is referred to as the rate of change. It can be interpreted as the change in the values of 𝒚 for every one-unit increase in the values of 𝒙.

When the rate of change is positive, the linear function is increasing. In other words, increasing indicates that as the 𝒙-value increases, so does the 𝒚-value.

When the rate of change is negative, the linear function is decreasing. Decreasing indicates that as the 𝒙-value increases, the 𝒚-value decreases.

The constant value is referred to as the initial value or 𝒚-intercept and can be interpreted as the value of 𝒚 when 𝒙 = 𝟎.