Linear Functions: Fresh Take

  • 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.

Writing the Point-Slope Form of a Linear Equation

The Main Idea

Point-slope form is especially useful when you know the slope and one point on the line, or when you know two points on the line.
The point-slope form is:

[latex]y - y_1 = m(x - x_1)[/latex]

where [latex]m[/latex] is the slope and [latex](x_1, y_1)[/latex] is a point on the line.

Point-slope form comes from the slope formula. Starting with [latex]m = \frac{y - y_1}{x - x_1}[/latex], we multiply both sides by [latex](x - x_1)[/latex] to get [latex]m(x - x_1) = y - y_1[/latex], which we rearrange as [latex]y - y_1 = m(x - x_1)[/latex].

Converting Between Forms

The Main Idea

Point-slope form and slope-intercept form describe the same line—they’re just different ways to write it. We can convert from one form to the other using basic algebra.
For example, if we have [latex]y - 4 = -\frac{1}{2}(x - 6)[/latex] in point-slope form:
[latex]\begin{align} y - 4 &= -\frac{1}{2}(x - 6) \\ y - 4 &= -\frac{1}{2}x + 3 && \text{distribute } -\frac{1}{2} \\ y &= -\frac{1}{2}x + 7 && \text{add 4 to both sides} \end{align}[/latex]
Both equations describe the same line!

 

Writing Equations Using a Point and the Slope

Question Help: Writing an Equation Given a Point and Slope

  1. Identify the slope [latex]m[/latex].
  2. Identify the coordinates [latex](x_1, y_1)[/latex] of the point.
  3. Substitute into point-slope form: [latex]y - y_1 = m(x - x_1)[/latex].
  4. If needed, convert to slope-intercept form by distributing and solving for [latex]y[/latex].

You can view the transcript for “Writing an equation using point slope form given a point and slope” here (opens in new window).

Writing Equations Using Two Points

When you know two points on a line but don’t know the slope, you can find the slope first, then use point-slope form.
The process:

  1. Calculate the slope using [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
  2. Choose either point to use as [latex](x_1, y_1)[/latex]
  3. Substitute into point-slope form

You can view the transcript for “Writing an equation using point slope form given two points” here (opens in new window).

When you have two points, you can use either one in the point-slope formula—you’ll get the same line! The equations might look different in point-slope form, but they’ll simplify to the same slope-intercept form.

Writing Equations from a Graph

Question Help: Writing an Equation from a Graph

  1. Identify two points on the line.
  2. Use the two points to calculate the slope.
  3. Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.
  4. Substitute the slope and y-intercept into slope-intercept form [latex]y = mx + b[/latex].

You can view the transcript for “Write a Slope Intercept Equation for a Line on a Graph” here (opens in new window).

Modeling Real-World Problems with Linear Functions

The Main Idea
Linear functions model many real-world situations where something changes at a constant rate:The key is identifying:

  • Initial value (the y-intercept [latex]b[/latex]): What you start with
  • Rate of change (the slope [latex]m[/latex]): How much changes per unit of time/input
Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, [latex]N[/latex], in his collection as a function of time, [latex]t[/latex] (in months). How many songs will he have in a year?

Working as an insurance salesperson, Ilya earns a base salary plus commission on each new policy. Last week he sold 3 policies and earned $760. The week before, he sold 5 policies and earned $920. Find an equation for [latex]I(n)[/latex], his weekly income as a function of policies sold, and interpret its meaning.

In linear models, the slope represents the rate of change (how much something increases per unit), and the y-intercept represents the starting value or fixed amount.

Horizontal and Vertical Lines

The Main Idea

Horizontal lines have a slope of 0. The y-value is constant for all x-values. Equation form: [latex]y = c[/latex] (where [latex]c[/latex] is a constant).

Vertical lines have an undefined slope. The x-value is constant for all y-values. Equation form: [latex]x = a[/latex] (where [latex]a[/latex] is a constant).

A horizontal line is a function (it passes the vertical line test), but a vertical line is NOT a function (it fails the vertical line test because one input maps to infinitely many outputs).

Parallel and Perpendicular Lines

The Main Idea

Parallel lines never intersect. They have the same slope but different y-intercepts.Perpendicular lines intersect at right angles (90°). Their slopes are negative reciprocals of each other.

You can view the transcript for “Determine if Lines are Parallel, Perpendicular or Neither” here (opens in new window).

Writing Equations of Parallel Lines

Question Help: Writing a Parallel Line Equation

  1. Find the slope of the given function.
  2. Use the same slope for the parallel line.
  3. Substitute the slope and the given point into either point-slope form or slope-intercept form.
  4. Simplify.

You can view the transcript for “Writing Equations of Parallel Lines Tutorial” here (opens in new window).

Writing Equations of Perpendicular Lines

The Main Idea
To write an equation of a line perpendicular to a given line:

  1. Find the slope of the given line
  2. Find the negative reciprocal of that slope
  3. Use this new slope and the given point to write the equation
  4. Simplify if needed

To find the negative reciprocal: flip the fraction and change the sign.

  • Slope 2 → negative reciprocal is [latex]-\frac{1}{2}[/latex]
  • Slope [latex]-\frac{3}{4}[/latex] → negative reciprocal is [latex]\frac{4}{3}[/latex]

Question Help: Writing a Perpendicular Line Equation

  1. Find the slope of the given function.
  2. Determine the negative reciprocal of the slope.
  3. Substitute the new slope and the values for [latex]x[/latex] and [latex]y[/latex] from the given point into [latex]y = mx + b[/latex].
  4. Solve for [latex]b[/latex].
  5. Write the equation for the line.

You can view the transcript for “Writing Equations of Perpendicular Lines Tutorial” here (opens in new window).