Linear Equations (Functions*)

Definitions:

  • y = mx + b (line equation)

  • m = (the slope of line)

  • b = (the number where line crosses the y-axis)

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    Use the following interactive example to explore how the graph of the line ( y = mx + b) changes as the values of the coefficients "m" and "b" change. (Note any useful observations in your journal.)

    *For functions substitute f(x) for y

     

     

    for a line "m" = (a constant)

     

    The slope is the constant rate of change of a linear equation. It can be thought of as the ratio of the vertical change to the horizontal change between two points on the graph of a line. If the two points are (x1, y1) and (x2, y2), then the vertical change is (y2 - y1) and the horizontal change is (x2 - x1).

     

    Hence we can use the formula to determine the slope of a line if we know two points on that line.

     

    See Figure 1 below.

    Figure 1

     

    Explore the following interactive example to see how the slope of a line is computed when two points are known. What observations can you make?

     

    When the slope of a linear equation is positive, the y-values increase as "x" increases. The graph is a line that rises from left to right. When the slope is negative, the y-values decrease as "x" increases, and the graph of the line falls as we go from left to right. If the slope of a linear equation is 0, then the "y" neither increases nor decreases, but remains constant. The graph of "y = a constant" is a horizontal line.

     

    Use the next interactive example to see how different values of the slope, m, affect the graph of the linear equation y = mx + b.

     

     

    Horizontal and Vertical Intercepts

    The points where a line crosses the vertical and horizontal axes are known as the vertical and horizontal intercepts.(These points are often referred to as the "x-intercept" and the "y-intercept").

    Given a linear equation y = mx + b,

  • The vertical intercept  (y-intercept) is found by evaluating the equation when the input variable, x, is "0" and is always the same as the constant "b". It can be thought of as the original value of the equation.

  • The horizontal intercept (x-intercept) is the value of the variable x when the y value is 0. It is found by solving the equation  0 = mx + b.

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    Interactive Example

    Now explore how the values of the y-intercept, "b", affect the graph of the linear equation

    y = mx + b.

     

     

    Algebraic Example

    Find the vertical and horizontal intercept of the linear equation

    f (x) = -7.2 x + 250

    Solution

     Since y = -7.2(0) + 250 = 250 for "x = 0", the vertical intercept is 250. This means that the graph of the linear function crosses the horizontal axis at the point (0, 250). Also notice that this is the value of b in the linear equation y = mx + b.

    To find the horizontal intercept we can replace "y" with "0" and solve the linear equation

    0 = -7.2 x + 250

    The solution is given below. 

    0 = -7.2 x + 250

    -250 = -7.2 x

    34.7 = x

    The "horizontal intercept" is "34.7". This is the point (34.7, 0) on the graph of the linear equation. A graph of the linear equation is shown in Figure 2.

    0 = -7.2 x + 250

    Figure 2