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Independent and dependent variables
Interpreting graphs Linear graphHere you will learn about straight line graphs including how to draw straight lines graphs in the form y=mx+b, using a table and from a pair of coordinates, and how to use the x and y intercepts to graph a line.
You will learn how to calculate the slope of a straight line including parallel and perpendicular lines, and how to quickly sketch a straight line graph with limited information.
Students first learn about linear equations in middle school when they work with the constant of proportionality. They build upon that knowledge as they progress through high school algebra.
The graph of a linear equation is a visual representation of a linear function. There are several ways you can express linear equations and graph linear equations. Letβs focus on the slope intercept form of a linear equation which is defined as:
y=mx+bWhere the slope (slant) of the line is represented by the variable, m and the y -intercept (point where the line crosses the y -axis) is represented by the variable, b.
Looking at the equation, y=-x+2 , you can define the slope of the line, m, to be the coefficient of the x term and the y -intercept, b, to be the constant.
In this case, m=-1 and b=2.
Letβs investigate strategies on how to use the equation of line to graph it on the coordinate plane.
Letβs use the equation below to graph the line using a table.
y=-x+2When using a table, you can select whatever values you would like for the x -coordinate because it is the independent variable to substitute into the equation and solve for y.
When selecting values for x, select numbers that are easy to calculate.
Once you complete the table, take each of the ordered pairs and plot them on the coordinate plane. If the points do not line up, then there could be a calculation error. So, be sure to go back and check your work.
Notice, the y -intercept is (0, 2) because it is the point where the line crosses the y -axis. Also, notice the point (2, 0). It is the x -intercept because it is the point where the line crosses the x -axis.
You can also see that from point to point there is a movement of 1 unit down and 1 unit right, which defines the slope of -1.
Step-by-step guide: How to find the slope of a line
Use this worksheet to check your 8th grade studentsβ understanding of graphing linear equations. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREEUse this worksheet to check your 8th grade studentsβ understanding of graphing linear equations. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREENow, letβs use the same linear equation to graph the line using the slope and the y -intercept.
y=-x+2First, identify the slope which in this case is -1. You know that the slope is -1 because it is the coefficient of the x term and it is the number in place of m.
The slope is defined as two movements the vertical movement over the horizontal movement or \cfrac{\text { rise }}{\text { run }}, \, so in this case, \cfrac{-1}{1} which indicates 1 unit down and 1 unit right.
Next, identify the y -intercept. In this case, the y -intercept is 2. You know that the y -intercept is 2 because it is the constant and the number in place of b.
Now letβs plot the line on the graph. Start by plotting the y -intercept. Since we know the y -intercept is 2 the actual ordered pair is (0, 2). Starting at that point, down 1 unit and to the right 1 unit.
Step-by-step guide: y = mx + b
Using the same linear equation, y=-x+2, letβs find the x -intercept and the y -intercept to graph the line on the coordinate plane.
Recall, the x -intercept is the point where the line intersects the x -axis and the y -intercept is the point where the line intersects the y -axis.
To find the x -intercept, substitute 0 into the equation for y and then solve for x. You can use this strategy because the ordered pair for a point on the x -axis has a 0 for the y -coordinate.
\begin{aligned} & y=-x+2 \\\\ & 0=-x+2 \\\\ & 0-2=-x+2-2 \\\\ & -2=-x \\\\ & \cfrac{-2}{-1}=\cfrac{-x}{-1} \\\\ & 2=x \text { or } x=2 \end{aligned}Since x = 2 , the x -intercept is 2 and it has an ordered pair of (2, 0).
To find the y -intercept, substitute the 0 into the equation for x and then solve for y. You can use this strategy because the ordered pair for a point on the y -axis has a 0 for the x -coordinate.
\begin{aligned} & y=-x+2 \\\\ & y=-(0)+2 \\\\ & y=2 \end{aligned}Since y = 2, the y -intercept is 2 and it has an ordered pair of (0, 2).
Plot the x -intercept and the y -intercept on the coordinate plane and connect the two points with a straight line. You can still see the slope to be 1 unit down and 1 unit right.
Step-by-step guide: How to find the y intercept
Letβs look at how to find the midpoint of a line segment and the distance between points on the coordinate plane.
The midpoint of a line segment is a point that lies exactly halfway between two points. It is the same distance from each endpoint of the straight line segment.
The endpoints of the line segment on the coordinate plane are (-2, 2) and (4, 6).
The formula you can use to find the midpoint is average of the x coordinates x\, (\text{midpoint})=\cfrac{x 1+x 2}{2} and the average of the y coordinates, y \, (\text{midpoint})=\cfrac{y 1+y 2}{2}
In this case, you can add the x coordinates together and then divide the sum by 2 and then do the same for the y coordinates.
x \, (\text{midpoint})=\cfrac{-2+4}{2}=\cfrac{2}{2}=1The x -coordinate of the midpoint is 1.
y \, (\text{midpoint})=\cfrac{2+6}{2}=\cfrac{8}{2}=4The y -coordinate of the midpoint is 4.
The midpoint is (1, 4)
You can find the distance between points on the coordinate plane or the length of the line segment.
Point 1\text{: } (-2, 2) where x1=-2 and y1=2
Point 2\text{: } (4, 6) where x2=4 and y2=6
\begin{aligned} & \text { distance }=\sqrt{(4-(-2))^2+(6-2)^2} \\\\ & \text { distance }=\sqrt{(4+2)^2+(4)^2} \\\\ & \text { distance }=\sqrt{6^2+4^2} \\\\ & \text { distance }=\sqrt{36+16} \\\\ & \text { distance }=\sqrt{52}=7.21 \end{aligned}Step-by-step guide: How to find the midpoint
Step-by-step guide: Distance formula
How does this relate to 8 th grade math and high school math?
There are several ways to graph linear equations. For more specific step-by-step guides, check out the graphing linear equations linked in the βWhat is graphing linear equations β section above or read through the examples below.
Make a table of values to sketch the line represented by y=-2x-3.
2Choose \bf{5} values for \textbf{x}.
3Substitute values of \textbf{x} into the equation to find \textbf{y} values.
4Plot the ordered pairs and draw a straight line through them.
Graph the straight line using the slope and y -intercept.
y=\cfrac{1}{3} \, x - 2Identify \textbf{m}, the slope and \textbf{b}, the \textbf{y} -intercept.
The slope, m, is the coefficient of the x term and the y -intercept, b, is the constant.
y=mx+b
In this case, m=\cfrac{1}{3} and b=-2
Plot the \textbf{y} -intercept.
Since the y -intercept is -2, the coordinate is (0, -2).
From the \textbf{y} -intercept move to the next point in the direction identified by the slope.
Draw a straight line through the \bf{2} points.
Graph the straight line using the slope and y -intercept.
y=-2 xIdentify \textbf{m}, the slope and \textbf{b}, the \textbf{y} -intercept.
The slope, m, is the coefficient of the x term and the y -intercept, b, is the constant.
y=mx+b
In this case, m=-2=\cfrac{-2}{1} and b=0.
Plot the \textbf{y} -intercept.
Since the y -intercept is 0, the coordinate is (0, 0).
From the \textbf{y} -intercept move to the next point in the direction identified by the slope.
Draw a straight line through the \bf{2} points.
Graph the straight line using the intercepts.
y=3x-6Find the \textbf{x} -intercept by substituting, \textbf{y} = \bf{0}, into the equation and solve for \textbf{x}.
y=3x-6, substitute y=0 into the equation.
x -intercept (2, 0)
Find the \textbf{y} -intercept by substituting, \textbf{x} = \bf{0}, into the equation and solve for \textbf{y}.
y=3x-6, substitute x=0 into the equation.
y -intercept (0, -6)
Plot the \textbf{x} -intercept and the \textbf{y} -intercept on the coordinate plane.
Draw a straight line through the \bf{2} intercepts.
Find the midpoint of the line segment that has endpoints (-4,3) and (0, -5).
Plot the line segment.
Use the midpoint formula to calculate the midpoint.
Point 1\text{: } (-4, 3) \; x1=-4 and y1=3
Point 2\text{: } (0, -5) \; x2=0 and y2=-5
Coordinates of midpoint (-2, -1)
Plot the midpoint of the line segment.
Find the distance between the points (7, 5) and (8, -2) on the coordinate.
Plot the points.
Use the distance formula to calculate the distance.
Point 1\text{: } (7, 5) \; x1=7 and y1=5
Point 2\text{: } (8,-2) \; x2=8 and y2=-2
Substitute the values into the formula to calculate the distance.
The distance between point 1 and point 2 is 7.07 units.
1) Which of these tables shows the correct values for the line y=6x-8?
Taking each of the values given for x, substitute them one at a time into the equation to find the value of y.
\begin{aligned} & x=-2 \\\\ & y=6(-2)-8 \\\\ & y=-20 \end{aligned}
\begin{aligned} & x=-1 \\\\ & y=6(-1)-8 \\\\ & y=-6-8 \\\\ & y=-14 \end{aligned}
\begin{aligned} & x=0 \\\\ & y=6(0)-8 \\\\ & y=-8 \end{aligned}
\begin{aligned} & x=1 \\\\ & y=6(1)-8 \\\\ & y=6-8 \\\\ & y=-2 \end{aligned}
\begin{aligned} & x=2 \\\\ & y=6(2)-8 \\\\ & y=12-8 \\\\ & y=4 \end{aligned}
\begin{aligned} & x=3 \\\\ & y=6(3)-8 \\\\ & y=18-8 \\\\ & y=10 \end{aligned}
2) Which graph represents the equation of a line, y=5x+3?
Using a table of values where the values of x are -2, -1, 0, 1.
From the graph, you can see the points (-2, -7), (-1, -2), (0, 3), (1, 8).
3) Using the slope and the y -intercept, which graph represents the equation
y=-\cfrac{1}{4} \, x?
The linear equation, y=-\cfrac{1}{4} \, x is in the form of y=mx+b where m=-\cfrac{1}{4} and b=0
So, the slope is \cfrac{-1}{4}=\cfrac{\text { down } 1 \text { unit }}{\text { right } 4 \text { units }} and the y -intercept is the point (0, 0).
4) Identify the slope and the y -intercept of the line represented by the equation, y=-5x+1.
m=-5 and b=-1
m=5 and b=1
m=-5 and b=-1
m=-5 and b=1
The slope intercept form of a linear equation is y=mx+b. The equation given is y=-5x+1 where m=-5 and b=1. So, the slope is -5 and the y -intercept is 1 which has the ordered pair (0, 1).
5) Find the x -intercept and the y -intercept of the line represented by the equation, y=4x-8.
x -intercept (2, 0) and Β y -intercept (0,8)
x -intercept (2, 0) and y -intercept (0,-8)
x -intercept (-2, 0) and y -intercept (0,-8)
x -intercept (-2,0) and y -intercept (0,8)
To find the x -intercept, substitute 0 for y and solve for x.
\begin{aligned} & y=4 x-8 \\\\ & 0=4 x-8 \\\\ & 0+8=4 x-8+8 \\\\ & 8=4 x \\\\ & \cfrac{8}{4}=\cfrac{4 x}{4} \\\\ & 2=x \end{aligned}
x -intercept (2, 0)
To find the y -intercept, you can substitute in a 0 for x and solve for y.
\begin{aligned} & y=4 x-8 \\\\ & y=4(0)-8 \\\\ & y=0-8 \\\\ & y=-8 \end{aligned}
y -intercept (0, -8)
6) Graph the line represented by the equation below using the x and the y intercepts.
2y+3x=12
To find the x -intercept, substitute 0 in for y and solve for x.
\begin{aligned} & 2 y+3 x=12 \\\\ & 2(0)+3 x=12 \\\\ & 0+3 x=12 \\\\ & 3 x=12 \\\\ & \cfrac{3 x}{3}=\cfrac{12}{3} \\\\ & x=4 \end{aligned}
x -intercept (4, 0)
To find the y -intercept, substitute 0 in for x and solve for y.
\begin{aligned} & 2 y+3 x=12 \\\\ & 2 y+3(0)=12 \\\\ & 2 y=12 \\\\ & \cfrac{2 y}{2}=\cfrac{12}{2} \\\\ & y=6 \end{aligned}
y -intercept (0, 6)
Yes, in algebra 1, you will learn three ways to write a linear equation β slope intercept form, standard form, and point-slope form.
Linear equations are classified as linear because the highest exponent or degree of the equation is 1.
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