Linear Functions: Learn It 1

  • Write the equation of a linear function given a point and a slope, two points, or a table of values.
  • Graph linear functions given any form of its equation.
  • Graph and write the equations of horizontal and vertical lines.
  • Write the equation of a line parallel or perpendicular to a given line.

If we want to write the equation of a linear function there are three forms we can choose from:

different forms of a linear function

  • Standard form: [latex]Ax+By=C[/latex]
  • Slope-intercept form: [latex]y=mx+b[/latex] where
    • [latex]m[/latex] is the slope, and
    • [latex]b[/latex] is the [latex]y[/latex]-intercept
  • Point-slope form: [latex]y-y_1=m(x-x_1)[/latex] where
    • [latex]m[/latex] is the slope, and
    • [latex](x_1,y_1)[/latex] is any point on the line.

Slope-Intercept Form

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as [latex]y=mx+b[/latex], where [latex]m=\text{ slope }[/latex] and [latex]b=y-\text{intercept}[/latex].

To calculate slope given two points, [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], [latex]m=\frac{\text{rise}}{\text{run}} = \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex].
Find the equation of the line for the graph below.This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3/2); (0, 2); (1, 5/2); and (2, 3).
Step 1: Calculate the Slope ([latex]m[/latex])The slope of a line is calculated using the formula [latex]m=\frac{\text{rise}}{\text{run}} = \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex].

  • Selecting two points from the graph: [latex](-2, 1)[/latex] and [latex](2, 3)[/latex].
  • Using these points, we can calculate the slope:

[latex]m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{3 - 1}{2 - (-2)} = \dfrac{2}{4} = \dfrac{1}{2}[/latex]

Step 2: Find the [latex]y[/latex]-intercept ([latex]b[/latex])

The [latex]y[/latex]-intercept is the value of [latex]y[/latex] when [latex]x=0[/latex]. From the graph, it is apparent that when [latex]x=0, y=2[/latex]. Therefore, [latex]b=2[/latex].

Step 3: Write the Equation

Now that we have the slope and [latex]y[/latex]-intercept, we can write the equation of the line:

[latex]y = \dfrac{1}{2}x+2[/latex]

Identify the slope and y-intercept given the equation [latex]y=-\frac{3}{4}x - 4[/latex].

slope-intercept form

The slope-intercept form of a line is written as:

[latex]y = mx+b[/latex]

where:

  • [latex]m[/latex] is the slope of the line, representing the rate of change or steepness of the line.
  • [latex]b[/latex] is the y-intercept, which is the point where the line crosses the y-axis. This value indicates where the line will pass through the y-axis when [latex]x=0[/latex].