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Graphs Using Slope-Intercept Form

Use the y-intercept and the 'rise over run' to graph a line

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Practice Graphs Using Slope-Intercept Form
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Graphing Linear Equations

Have you ever been on a hiking trip where you needed to pack your own food? Take a look at this dilemma.

The rainforest was such a popular topic of discussion that Mr. Thomas let the students talk about it all week. They loved discussing all of the things that they had seen. One day, they began talking about the scientists and all of the things that they had to carry with them into the rainforest.

“You know, they couldn’t exactly run out to the store to pick something up,” Casey commented.

“Or call for a pizza!” Susan said.

Mr. Thomas once again seized a great opportunity to write the following problem on the board.

A group of backpackers leaves with 84lbs. of food. They plan to eat 11lbs. per day. Use an equation to show on a graph how much food they should have after each day. How long should their food last?

To work on this problem, you will need to know about slope. The food changes per day based on how much the backpackers eat. This Concept will teach you all that you need to know to work on this problem.

Guidance

The slope-intercept form of an equation, y=mx+b\begin{align*}y=mx+b\end{align*}, is very helpful when you need to find the slope and y\begin{align*}y\end{align*}-intercept. Using this form, graphing is going to be easy, too. Since we know the slope and we know the y\begin{align*}y\end{align*}-intercept, then instead of using a table of values, we can plot the y\begin{align*}y\end{align*}-intercept on the coordinate plane and find our next point using the slope.

For any equation written in the form y=mx+b\begin{align*}y=mx+b\end{align*}, m\begin{align*}m\end{align*} is the slope and b\begin{align*}b\end{align*} is the y\begin{align*}y\end{align*}-intercept.

For that reason, y=mx+b\begin{align*}y=mx+b\end{align*} is called the slope-intercept form.

Using the properties of equations, you can write any equation in this form.

Because we can use slope – intercept form, we can rewrite equations in standard form into slope – intercept form. Then we can easily determine the slope and y\begin{align*}y\end{align*} – intercept of each equation.

Now let's use the slope and the y-intercept to graph equations.

Graph the line that goes with the equation y=x+5\begin{align*}y=-x+5\end{align*}

First, we can determine that the slope is -1 and the y\begin{align*}y \end{align*} – intercept is 5. Next, we can graph this line using these two pieces of information.

We can also graph lines in a different form. First, we will need to rewrite them into slope – intercept form. Then we can graph the equation.

Take a look at this situation.

Graph the line 3x+y=9\begin{align*}3x+y=9\end{align*}

First, we rewrite this equation from standard form to slope – intercept form. We do this by using inverse operations.

3x3x+yy=3x+9=3x+9

Now we know that the slope is -3 and the y\begin{align*}y \end{align*} – intercept is 9. Next, we can graph the equation of the line.

Use what you have learned to answer each question.

Example A

True or false. The slope of a horizontal line is greater than the slope of a vertical line.

Solution: False.

Example B

Identify the slope in the following equation.

y=2x+7\begin{align*}y=-2x+7\end{align*}

Solution: 2\begin{align*}-2\end{align*}

Example C

Identify the slope and the y-intercept of this equation.

3x3y=18\begin{align*}-3x-3y=18\end{align*}

Solution: slope=1,yintercept=6\begin{align*}slope = -1, y-intercept = -6\end{align*}

Now let's go back to the dilemma at the beginning of the Concept.

First, we need to write an equation to represent the food. Use the equation f=11d+84\begin{align*}f=-11d+84\end{align*} where f\begin{align*}f\end{align*} is the food they have remaining and d\begin{align*}d\end{align*} is the number of days that have been gone.

The graph shows that they have enough food for a little more than seven days.

Vocabulary

Slope – Intercept Form
the form of an equation y=mx+b\begin{align*}y=mx+b\end{align*}
Standard Form
the form of an equation Ax+By=C\begin{align*}Ax+By=C\end{align*}
Slope
the steepness of the line, calculated by the ratio of rise over run.
y\begin{align*}y \end{align*} – Intercept
the point where a line crosses the y\begin{align*}y\end{align*}axis.

Guided Practice

Here is one for you to try on your own.

A store employee gets paid $12.50 per hour plus a weekly bonus of$50 for punctuality. Assuming the employee is on time every day, graph her wages earned. How much would she earn for working 10 hours? 20 hours? 30 hours?

Solution

First, write an equation. We could use the equation w=12.5h+50\begin{align*}w=12.5h+50\end{align*}.

Next, we can graph the equation by using the slope and the y-intercept.

The graph shows that wages for 10 hours would be $175, for 20 hours$300 and for 30 hours 425. Video Review Practice Directions: Use what you have learned to complete each task. 2x+2y=8\begin{align*}2x+2y=8\end{align*} 1. Write this equation in slope – intercept form. 2. What is the slope? 3. What is the y\begin{align*}y\end{align*} – intercept? 4. Graph the equation. 3x+6y=2\begin{align*}3x+6y=2\end{align*} 1. Write the following equation in slope – intercept form. 2. What is the slope? 3. What is the y\begin{align*}y\end{align*} – intercept? 4. Graph the equation. Directions: Use what you have learned to solve each problem. Miguel wants to save47 for a video game. He received $20 as a gift and gets$4 per week for allowance.

1. Write an equation in slope-intercept form that represents this situation.
2. How long will it take him to save enough money?

y=.8x+3\begin{align*}y=.8x+3\end{align*}

1. What is the slope of this line?
2. Which form is the equation written in?
3. What is the y-intercept of this line?
4. What is the graph of this line?
5. Is this a linear graph?How can you tell?

Vocabulary Language: English

$y-$intercept

$y-$intercept

A $y-$intercept is a location where a graph crosses the $y-$axis. As a coordinate pair, this point will always have the form $(0, y)$.
Cartesian Plane

Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.
linear equation

linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.
Linear Function

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.
Slope

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Slope-Intercept Form

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b,$ where $m$ is the slope and $b$ is the $y-$intercept.