7.7.a

A linear function is a function which can be written in the form y = mx + b, where m is the slope and b is where the line crosses the y-axis.

The point where a line crosses the y-axis is called the y-intercept.
At this point, the x-value is 0.

The table, graph, and equation each represent the same line.
The y-intercept of the line is (0, 1).

x	y
-2	-1
-1	0
0	1
1	2

y = x + 1

The slope of a line is also called the rate of change. Given two points on a line, (x_₁, y_₁) and (x_₂, y_₂), the slope is the ratio of the change in y to the change in x.

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

In the equation above, y = x + 1, the slope is the coefficient on x, which is 1.

In the table above, choose any two points to find the slope.
The calculation below shows finding the slope using the points (0, 1) and (1, 2).

\begin{array}{rcl} m & = & \frac{2 - 1}{1 - 0} \\ = & \frac{1}{1} \\ = & 1 \end{array}

In the graph above, choose any two points to find the slope.
The calculation below shows finding the slope using the points (1, 2) and (2, 3).

\begin{array}{rcl} m & = & \frac{3 - 2}{2 - 1} \\ = & \frac{1}{1} \\ = & 1 \end{array}

Linear Representations

Texas Essential Knowledge and Skills (TEKS): 7.7.a

represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.

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