Linear Functions: Basics and Applications

Explanation

Definition

Linear functions describe a straight relationship between two variables. They are also referred to as linear equations. The basic form shows that the slope is constant. With the help of the slope triangle, one can easily calculate the slope m.

y = m \cdot x + b

Procedure

Scheme

Function term: Determine the expression of a linear function by plugging the slope m and the y-intercept b into the basic form.

y = m \cdot x + b

Given are m=2 and b=3. Insert this into the formula.

y = 2x + 3

Function graph: Draw the line by taking the y-intercept as the starting point and using the slope m to determine another point.

Draw the function y = -x + 2.

The y-intercept is at (0,2). With m=-1 another point arises, for example (1,1).

Slope: Determine the slope m by dividing the difference of the y-values by the difference of the x-values.

m = \frac{y_2 - y_1}{x_2 - x_1}

For the points (1,4) and (3,8): m = \tfrac{8-4}{3-1} = 2.

Intercepts: The y-intercept is the value b in the function. For the x-intercept you set y = 0 and solve for x on.

x = -\frac{b}{m}

Given is y = 2x + 3: Set 0 = 2x + 3 and get x = -\tfrac{3}{2}.

Misunderstandings

Common Errors

★It is often forgotten to recognize the y-intercept as b in the equation.
★It is often forgotten to construct the slope triangle correctly to calculate the slope m.
★It is often forgotten to verify the x-intercept by setting y = 0.

Problems

1 / 3

Determine the function term of a linear function with

m=1, b=2

2 / 3

Draw the graph of the function

y = -3x + 1

3 / 3

Calculate the slope and the function term of the line through the points

(2,3) and (-1,-6)

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