Slope-Intercept Form of an Equation © 2002 by Shawna Haider

In the section 2.1 the examples suggest that for any number m, the graph of y = mx is a straight line passing through the origin. What happens when we add a number b on the right side and graph the equation y = mx + b. Let’s try some examples.

Graph y = 2x and y = 2x + 3 xyy y = 2xy = 2x Solution: We first make a table of solutions of both equations. 3 units up Note that the graph y = 2x + 3 passes through the point (0, 3)

Graph y = 1/3x and y = 1/3x - 2 xyy y = 1/3xy = 1/3x Solution: We first make a table of solutions of both equations. 2 units down Note that the graph y = 1/3x - 2 passes through the point (0, -2)

The graph of any line written in the form y = mx + b passes through point (0, b). The point (0, b) is called the y-intercept For each equation, find the y- intercept. y = 5.3x - 12 y = -5x + 7 The y – intercept is (0, 7) The y – intercept is (0, -12)

SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is given by

Slope Problem Examples Determine a value for x such that the line through the points has the given slope. Let's use the slope formula and plug in what we know. (x1,y1)(x1,y1)(x2,y2)(x2,y2) You can cross-multiply to find a fraction-free equation for x to solve.

Example when you have a point and the slope A point on a line and the slope of the line are given. Find two additional points on the line. 0 Remember that slope is the change in y over the change in x. The slope is 2 which can be made into the fraction (0,-3) So this point is on the line also. You can see that this point is changing (adding) 2 to the y value of the given point and changing (adding) 1 to the x value To find another point on the line, repeat this process with your new point (0,-3) +1+2 (1,-1) (-1,5)

Determine the slope and the y-intercept of the line given by Slope-intercept form slope y intercept m = -1/3 SLOPE b = 2 y - intercept

y intercept slope Example of given an equation, find the slope and y intercept Find the slope and y intercept of the given equation and graph it. Now plot the y intercept From the y intercept, count the slope Change in y Change in x Now that you have 2 points you can draw the line

y intercept slope Example of given an equation, find the slope and y intercept Find the slope and y intercept of the given equation and graph it. First let's get this in slope- intercept form by solving for y. -3x Now plot the y intercept From the y intercept, count the slope Change in y Change in x Now that you have 2 points you can draw the line

These will be linear models. When you read the problem look for two different variables that can be paired together in ordered pairs. Example: The total sales for a new sportswear store were $150,000 for the third year and $250,000 for the fifth year. Find a linear model to represent the data. Estimate the total sales for the sixth year. Total sales depend on which year so let’s make ordered pairs (x, y) with x being the year and y being the total sales for that year. (3, 150,000) (5, 250,000) We now have two points and can determine a line that contains these the points (on the next screen)

We’ll want to use the point-slope equation (3, 150,000)(5, 250,000) First we need the slope Now we have all the pieces we need so we can plug them in We now have an equation to estimate sales in any given year. To estimate sales in the sixth year plug a 6 in for x.

Slope-Intercept Form Useful for graphing since m is the slope and b is the y-intercept Point-Slope Form Use this form when you know a point on the line and the slope Also can use this version if you have two points on the line because you can first find the slope using the slope formula and then use one of the points and the slope in this equation. General Form Commonly used to write linear equation problems or express answers

Remember parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same. Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.

Example of how to find x and y intercepts to graph a line The x-intercept is where a line crosses the x axis (6,0) (-1,0) (2,0) What is the common thing you notice about the x-intercepts of these lines? The y value of the point where they cross the axis will always be 0 To find the x-intercept when we have an equation then, we will want the y value to be zero.

Now let's see how to find the y-intercept The y-intercept is where a line crosses the y axis (0,4) (0,1) (0,5) What is the common thing you notice about the y-intercepts of these lines? The x value of the point where they cross the axis will always be 0 To find the y-intercept when we have an equation then, we will want the x value to be zero.

Let's look at the equation 2x – 3y = 12 Find the x-intercept. We'll do this by plugging 0 in for y 2x – 3(0) = 12 Now solve for x. 2x = x = 6 So the place where this line crosses the x axis is (6, 0)

2x – 3y = 12 Find the y-intercept. We'll do this by plugging 0 in for x 2(0) – 3y = 12 Now solve for y. -3y = y = - 4 So the place where this line crosses the y axis is (0, -4) We now have enough information to graph the line by joining up these points (6,0) (0,- 4)

Graphing Horizontal and Vertical Lines

Horizontal Lines The slope of a horizontal line 0 The graph of any function of the form f(x) = b or y = b is a horizontal line that crosses the y-axis at (0, b).

Vertical Lines

The slope of a Vertical line is undefined The graph of any graph in the form of x = b is a vertical line that crosses the x-axis at (b, 0).