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Functions and Graphs Help (page 3)

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Rise Over Run

Okay, now you know that slope means the steepness of a line. In the equation y = 2x + 3, the slope of the line is 2. What does it mean when you have a slope of 2? Slope is defined as the rise of the line over the run of the line. If the slope is 2, this means , so the rise is 2 and the run is 1.

If the slope of a line is , the rise is 2 and the run is 3. What do rise and run mean? Rise is the vertical change, and run is the horizontal change. To graph a line passing through the origin with a slope of , start at the origin. The rise is 2, so from the origin, go up 2 and to the right 3. Then draw a line from the origin to the endpoint. The line you have drawn has a slope of .

Now draw a line with a slope of . Start at the origin. Go down 3 units because you have a negative slope. Then go right 4 units. Finally, draw a line from the origin to the endpoint. These two lines appear on the same graph that follows.

TIP: Always look at all coordinate planes carefully to see what scale each is drawn to—some may have increments of one, while others may have increments of one-half or ten.

Let's try another one. To graph an equation like y = x + 1, you can use the slope and y-intercept. The first step is to figure out what the slope is. The slope is the number in front of x, which means in this case that it is 1. What is the y-intercept? It is also 1. To graph the equation, your starting point will be the y-intercept, which is 1. From the y-intercept, use the slope, which is also 1, or . The slope tells you to go up 1 and to the right 1. A line is drawn from the y-intercept to the endpoint (1,2). You can extend this line and draw arrows on each end to show that the line extends infinitely.

Find the Slope of an Equation - Getting the Right Form

What if the equation is not in slope-intercept form? Simple! All you need to do is change the equation to slope-intercept form. How? Slope-intercept form is y = form, so your strategy is to get the y on a side by itself.

An equation needs to be in slope-intercept form, or y = form (y = mx + b), before you can graph the equation with a pencil and graph paper. Also, if you use a graphing calculator to graph a linear equation, the equation needs to be in y = form before it can be entered into the calculator.

Example

 2x + y = 5 Subtract 2x from both sides of the equation. 2x – 2x + y = 5 – 2x Simplify. y = 5 – 2x Rearrange the equation so the x term is first. y = –2x + 5

Find the Slope of an Equation - Using the Commutative Property

There is a mathematical rule called the commutative property that lets you change the order of numbers or terms when you add or multiply. You want the preceding equation in the form y = mx+ b, so the order of the 5 and the –2x needs to be changed after getting the y on a side by itself. When you move a term, be sure to take the sign of the term with it. For example, the 5 was a positive number in the original order. It remains a positive number when you move it.

Example

 2x + 3y = 9 Subtract 2x from both sides of the equation. 2x – 2x + 3y = 9 – 2x Simplify 3y = 9 – 2x Use the commutative property. 3y = –2x + 9 Divide both sides by 3. Simplify both sides of the equation.

Tip: Look at the last two steps of the 2x + 3y = 9 example. Because and x name the same number, you may see it written either way.

Example

 –3x + 2y = 10 Add 3x to both sides of the equation. –3x + 3x + 2y = 10 + 3x Simplify. 2y = 10 + 3x Use the commutative property. 2y = 3x + 10 Divide both sides of the equation by 2. Simplify both sides of the equation.

Linear Inequalities

The Deal with Linear Inequalities

A linear inequality has two variables just like a linear equation. The inequality 2x + y < 1 is a linear inequality with two variables. You can draw on what you already know to graph linear inequalities. A linear equation graphs into a line. A linear inequality has two parts: a line and a shaded area.

When you graphed linear equations, your first step was to put the equation into y = form. Do the same with the linear inequality. The commutative property lets you change the order of numbers or terms when you add or multiply. When you move a term, be sure to take the sign of the term with it.

Tip: If the inequality symbol is < or >, the boundary line will be dotted. If the inequality symbol is or , the boundary line will be solid.

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