Day 6 – Vertical & Horizontal Lines

Example 1 Compare the graphs of 𝑦=4 𝑎𝑛𝑑 𝑥=4. The graph 𝑦=4 consists of all points with a y-coordinate of 4. the graph 𝑥= 4 consists of all points with an x- coordinate of 4.

The equation 𝑦=4 is a horizontal line 4 units above the origin.

𝑥=4 The equation x=4 is a vertical line 4 units to the right of the origin.

Example 2 Find the slope a) The graph of 𝑦=5 has the same value of y for any value of x. Use any two points with a y-coordinate of 5 to find the slope. The points (3, 5) and (7, 5) will work. 𝑠𝑙𝑜𝑝𝑒= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = 5−5 7−3 = 0 4 =0

Example 2 Find the slope b) The graph of 𝑥=−3 has the same value of 𝑥 for any value of y. Use any two points with any two points with an x-coordinate of -3 to find the slope. The points (−3, 3) and (−3, 7) will work. 𝑠𝑙𝑜𝑝𝑒= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = 7−3 (−3)−(−3) = 4 0

The run is 0. since dividing by 0 is impossible, the slope is undefined. The graph of an equation in the form 𝑥=𝑎 is a vertical line that crosses the 𝑥−𝑎𝑥𝑖𝑠 where x equals a. The equation cannot be written in slope- intercept form, but it can be written in standard form. 1𝑥+0𝑦=𝑎

Horizontal and Vertical Lines The equation for a horizontal line is written in the form 𝑦=𝑏. The slope is 0. The equation for a vertical line is written in the form 𝑥=𝑎. The slope in undefined.

Example 3 Compare the graphs of 𝑦=3𝑥 𝑎𝑛𝑑 𝑥=3𝑦. Write the equations in slope-intercept form. Graph the lines. The equation 𝑦=3𝑥 is in slope-intercept form, and its graph is a line with slope 3 and y-intercept 0.

Example 3 Change 𝑥=3𝑦 to slope-intercept form by solving for y. Since 𝑥=3𝑦 is equivalent to 3𝑦=𝑥, 3𝑦=𝑥→𝑦= 𝑥 3 →𝑦= 1 3 𝑥+0 The equation 𝑥=3𝑦 is written 𝑦= 1 3 𝑥+0 in slope-intercept form. Its graph is a line with slope 1 3 and y-intercept 0.

𝑦=3𝑥 𝑥=3y or 𝑦= 1 3 𝑥 Recall that numbers are reciprocals when their product equals 1. since 3∙ 1 3 =1, the slopes of 𝑦=3𝑥 and 𝑦= 1 3 𝑥 are reciprocals. The graphs of 𝑦=3𝑥 and 𝑥=3𝑦 are lines with reciprocals slope and a common points at the origin, (0, 0).

Draw graph of 𝑦=𝑥. On the same set of axes, draw the graph of 𝑦=2𝑥 and 𝑥=2𝑦. Try other pairs of lines in the form 𝑦=𝑚𝑥 and 𝑥=𝑚𝑦 with different 𝑚−𝑣𝑎𝑙𝑢𝑒𝑠 . What relationships do you see? These pairs of line will have reciprocal slope and pass through the common point (0, 0).