Graphing Linear Functions - Algebra 2
Card 0 of 92
What is the equation of the line displayed above?
What is the equation of the line displayed above?
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The equation of a line is 
, with m being the slope of the line, and b being the y-intercept. The y intercept of the line is at 
, so 
.
The x-intercept is at 
, the equation becomes 
, simplification yields 
The equation of a line is
The x-intercept is at
Determine where the graphs of the following equations will intersect.
Determine where the graphs of the following equations will intersect.
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We can solve the system of equations using the substitution method.
Solve for 
in the second equation.
Substitute this value of 
into the first equation.
Now we can solve for 
.
Solve for 
using the first equation with this new value of 
.
The solution is the ordered pair 
.
We can solve the system of equations using the substitution method.
Solve for
Substitute this value of
Now we can solve for
Solve for
The solution is the ordered pair
Which of the following graphs correctly depicts the graph of the inequality 
Which of the following graphs correctly depicts the graph of the inequality
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Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to 
and the y-intercept is equal to 
.
All of the graphs depict a line with slope of 
and y-intercept 
. Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin 
. If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to
All of the graphs depict a line with slope of
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?
Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?
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A horizontal line has equation 
for some value of 
; since the line goes through a point with 
-coordinate 3, the line is 
. Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is 
.
A vertical line has equation 
for some value of 
; since the line goes through a point with 
-coordinate 4, the line is 
. Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is 
.
Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is 
.
A horizontal line has equation
A vertical line has equation
Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is
Which of the following inequalities is graphed above?
Which of the following inequalities is graphed above?
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First, we determine the equation of the boundary line. This line includes points 
and 
, so the slope can be calculated as follows:
Since we also know the 
-intercept is 
, we can substitute 
in the slope-intercept form to obtain equation of the boundary:
The boundary is excluded, as is indicated by the line being dashed, so the equality symbol is replaced by either 
or 
. To find out which one, we can test a point in the solution set - we will choose 
:
1 is greater than 0 so the correct symbol is 
The inequality is 
First, we determine the equation of the boundary line. This line includes points
Since we also know the
The boundary is excluded, as is indicated by the line being dashed, so the equality symbol is replaced by either
1 is greater than 0 so the correct symbol is
The inequality is
Which of the following inequalities is graphed above?
Which of the following inequalities is graphed above?
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First, we determine the equation of the boundary line. This line includes points 
and 
, so the slope can be calculated as follows:
Since we also know the 
-intercept is 
, we can substitute 
in the slope-intercept form to obtain the equation of the boundary line:
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either 
or 
. To find out which one, we can test a point in the solution set - for ease, we will choose 
:
0 is less than 3 so the correct symbol is 
.
The inequality is 
.
First, we determine the equation of the boundary line. This line includes points
Since we also know the
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either
0 is less than 3 so the correct symbol is
The inequality is
Which equation best matches the graph of the line shown above?
Which equation best matches the graph of the line shown above?
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To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.
Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.
Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:
which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:
To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.
Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.
Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:
which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:
What is the equation of the above line?
What is the equation of the above line?
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The equation of a line is 
with m being the slope and b being the y intercept. The y-intercept is at 
, so 
. The x-intercept is 
, so after plugging in the equation becomes 
, simplifying to 
.
The equation of a line is
Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the 
-axis, where would its 
-intercept be?
Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the
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First, we need to find the slope of the line.
In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown 
.
We can use this to find the 
-intercept 
using the slope formula as follows:
The lower left point has coordinates 
. Therefore, we can set up and solve for 
in this slope formula, setting 
:
First, we need to find the slope of the line.
In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown
We can use this to find the
The lower left point has coordinates
Line 
includes the points 
and 
. Line 
includes the points 
and 
. Which of the following statements is true of these lines?
Line
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We calculate the slopes of the lines using the slope formula.
The slope of line 
is
The slope of line 
is
The lines have the same slope, making them either parallel or identical.
Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form 
, where 
is the 
-coordinate of each point on the line. Therefore, line 
and line 
have equations 
and 
.This makes them parallel lines.
We calculate the slopes of the lines using the slope formula.
The slope of line
The slope of line
The lines have the same slope, making them either parallel or identical.
Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form
An individual's maximum heart rate can be found by subtracting his or her age from 
. Which graph correctly expresses this relationship between years of age and maximum heart rate?
An individual's maximum heart rate can be found by subtracting his or her age from
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In 
form, where y = maximum heart rate and x = age, we can express the relationship as:
We are looking for a graph with a slope of -1 and a y-intercept of 220.
The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.
In
We are looking for a graph with a slope of -1 and a y-intercept of 220.
The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.
Which of the following is the function graphed below?
Which of the following is the function graphed below?
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This function is linear (a line), so we must remember that we can represent lines algebraically using y=mx+b, where m is the slope and b is the y-intercept.
Looking at the graph, we can tell immediately that the y-intercept is -5, because the line crosses(intercepts) the y-axis at -5.
To find the slope, we need two points, 
and the following formula:
.
For the sake of the example, choose (0,-5) and (2,-1). We can see that the graph clearly passes through each of these points. Any two points will do, however. Substituting each of the values into the slope formula yields m=2.
Thus, our final answer is 
This function is linear (a line), so we must remember that we can represent lines algebraically using y=mx+b, where m is the slope and b is the y-intercept.
Looking at the graph, we can tell immediately that the y-intercept is -5, because the line crosses(intercepts) the y-axis at -5.
To find the slope, we need two points,
For the sake of the example, choose (0,-5) and (2,-1). We can see that the graph clearly passes through each of these points. Any two points will do, however. Substituting each of the values into the slope formula yields m=2.
Thus, our final answer is
Where does 
cross the 
axis?
Where does
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To find where this equation crosses the 
axis or its 
-intercept, change the equation into slope intercept form.
Subtract to isolate 
:
Divide both sides by 
to completely isolate 
:
This form is the slope intercept form 
where 
is the slope of the line and 
is the 
-intercept.
To find where this equation crosses the
Subtract to isolate
Divide both sides by
This form is the slope intercept form
Select the equation of the line perpendicular to the graph of 
.
Select the equation of the line perpendicular to the graph of
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Lines are perpendicular when their slopes are the negative recicprocals of each other such as 
. To find the slope of our equation we must change it to slope y-intercept form.
Subtract the x variable from both sides:
Divide by 4 to isolate y:
The negative reciprocal of the above slope: 
. The only equation with this slope is 
.
Lines are perpendicular when their slopes are the negative recicprocals of each other such as
Subtract the x variable from both sides:
Divide by 4 to isolate y:
The negative reciprocal of the above slope:
Find the 
-intercepts and the 
-intercepts of the equation.
Find the
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To find the x-intercepts, remember that the line is crossing the x-axis, and that y=0 when the line crosses the x-axis.
So plug in y=0 into the equation above.
To find the y-intercepts, remember that the line is crossing the y-axis, and that x=0 when the line crosses the x-axis.
So plug in x=0 into the equation above.
To find the x-intercepts, remember that the line is crossing the x-axis, and that y=0 when the line crosses the x-axis.
So plug in y=0 into the equation above.
To find the y-intercepts, remember that the line is crossing the y-axis, and that x=0 when the line crosses the x-axis.
So plug in x=0 into the equation above.
Find the slope of the line that passes through the pair of points. Express the fraction in simplest form.
and 
Find the slope of the line that passes through the pair of points. Express the fraction in simplest form.
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Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.
Therefore the formula for slope is as follows.
Plugging in our given points
and 
Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.
Therefore the formula for slope is as follows.
Plugging in our given points
Which of the following equations passes through 
and is parallel to 
.
Which of the following equations passes through
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Since the line goes through 
we know that 
is the y-intercept.
Since we are looking for parallel lines, we need to write the equation of a line that has the same slope as the original, which is 
.
Slope-intercept form equation is 
, where 
is the slope and 
is the y-intercept.
Therefore,
.
Since the line goes through
Since we are looking for parallel lines, we need to write the equation of a line that has the same slope as the original, which is
Slope-intercept form equation is
Therefore,
Write an equation of the line passing through 
and 
in slope-intercept form.
Reminder: Slope-Intercept form is 
, where 
is the slope and 
is the y-intercept.
Write an equation of the line passing through
Reminder: Slope-Intercept form is
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Step 1: Find the Slope
Step 2: Find the y-intercept
Use the slope and a point in the original y-intercept
Step 3: Write your equation
Step 1: Find the Slope
Step 2: Find the y-intercept
Use the slope and a point in the original y-intercept
Step 3: Write your equation
Find the slope-intercept form of an equation of the line that has a slope of 
and passes through 
.
Find the slope-intercept form of an equation of the line that has a slope of
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Since we know the slope and we know a point on the line we can use those two piece of information to find the y-intercept.
Since we know the slope and we know a point on the line we can use those two piece of information to find the y-intercept.
Determine the slope of a line that has points 
and 
.
Determine the slope of a line that has points
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Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.
Therefore the formula for slope is as follows.
Plugging in our given points
and 
,
.
Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.
Therefore the formula for slope is as follows.
Plugging in our given points