Sketch and write the equation for each line

To find the equation of a graphed line, find the y-intercept and the slope in order to write the equation in y-intercept (y=mx+b) form. Slope is the change in y over the change in x. Find two points on the line and draw a slope triangle connecting the two points. Label the legs of the triangle. This will tell you the rise (change in y, numerator) value and the run (change in x, denominator) value. The y-intercept comes from the point where the line passes the y-axis. The y-value of the coordinate is the y-intercept, or the "b" value. Plug in the slope, m, and the y-intercept, b, to the slope-intercept form of the line.

This type of problem is all over the Algebra 1 course. By type of problem I mean where you are given a graph and you are asked to write its equation. So when you're doing that there's a lot of different ways to approach it. My personal favorite way is to find the slope and the y intercept. Especially in graphs like this where those things kind of jump out at you. Here's what I mean.

For the equation of a line I'm thinking y equals mx plus b form. I need the slope and the y intercept. This y letter and that x letter are going to stay in my equation, so let's go ahead and feel in the blanks.

My slope number can be found by the change in y on top of the change in x. One way to do it is to find any two points that the line goes through exactly and draw a slope triangle. So let's see, this line goes exactly through that point and that point right there. So I want to draw the slope triangle which connects them, all right and then where I label the sides that's going to tell me my rise and my run and then slope is vertical change on top of horizontal change. So in my case my slope is going to be +3/2. I'm almost done. +3/2, sweet, not so bad. The last thing I need from my y equals mx plus b equation is the y intercept.

Remember the y is the vertical axis so the y intercept comes from where your graphed line crosses that vertical axis. There it is right there the coordinates are 0 for x, 3 for y. The 3 is the important piece that's going to be my b value for my y intercept. The equation for the line of will be y equals 3/2x plus 3.

That's it. When you have a graph like that and especially when it's on graph paper like a graph grid it's really easy to find the equation in Slope-Intercept form. The first thing you do is find the slope second thing you do is find the y intercept and then just plug them in.

Line Equations - How to Draw a Line

(How to draw a line, given its equation)

We now learn how to draw a line, given its line equation.

By the end of this section we'll know how to show, for example, that the line with equation: \[y = x-4\] is the line shown here:

We start by learning the method, which consists of finding two points through which the line passes.
We then illustrate this technique with a tutorial.
Finally we consolidate what we have learnt by working through some exercise questions, each of which has both an answer key as well as a detailed video solution.

Method

Given a line equation \(y = mx + c\), to draw the line we need two points through which the line passes.

• Point 1: the \(y\)-intercept: this is the point at which the line cuts the \(y\)-axis, its coordinates are always: \[\begin{pmatrix}0,c\end{pmatrix}\] where \(c\) can be read directly from the equation \(y=mx+c\).
• Point 2: the \(x\)-intercept: this is the point at which the line cuts the \(x\)-axis.
  To find it we follow two steps:
  • Step 1: replace \(y\) by \(0\) in the line equation \(y=mx+c\), so that we have: \[0 = mx+c\]
  • Step 2: solve \(0 = mx+c\) for \(x\).
    The value of \(x\) found will be the \(x\) coordinate of the \(x\)-intercept.

Note: This method is best illustrated with an example. Watch the tutorial, below, in which we see how this method works.

Tutorial

In the following tutorial we learn how to draw a line, given its equation, by finding both its \(y\)-intercept and its \(x\)-intercept.

We learn the method by drawing the line with equation: \[y = 2x - 6\]

Exercise

1. Draw the line with equation: \[y = 2x-8\]
2. Draw the line with equation: \[y = -3x+12\]
3. Draw the line with equation: \[y = x-5\]
4. Draw the line with equation: \[y = \frac{x}{2}-3\]
Draw the line with equation: \[y = -2x+6\]

Answers Without Working

Answers with Working

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