Transformations of Linear Functions - Algebra II
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Which line would never intersect a line with the slope 
?
Which line would never intersect a line with the slope
This question is very simple once you realize that a line that will never intersect another line must have the same slope (parallel lines will never intersect). Therefore you must look for the choice that has a slope of 
. Each answer can be converted to the form 
or by knowing that in the equation 
, the slope of the line is simply 
. In the correct answer, 
, the slope would be 
, which simplfies to 
.
*Note* the y-intercept is irrelevant to finding the correct answer.
This question is very simple once you realize that a line that will never intersect another line must have the same slope (parallel lines will never intersect). Therefore you must look for the choice that has a slope of
*Note* the y-intercept is irrelevant to finding the correct answer.
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What is the equation of the line that intersects the point 
and 
?
What is the equation of the line that intersects the point
We are only given the points the line intersects. This can be used to find the slope of the line, knowing that slope is rise/run, or change in 
/change in 
or by the formula,
.
By substituting, we get
for the slope.
To find the 
intercept, we can use the equation 
, where 
---> 
.
Since both given points are on the line, either can be used to solve for 
:
--> 
--> 
We are only given the points the line intersects. This can be used to find the slope of the line, knowing that slope is rise/run, or change in
By substituting, we get
To find the
Since both given points are on the line, either can be used to solve for
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Which line is perpendicular to the line 
?
Which line is perpendicular to the line
Lines that are perpendicular have negative reciprocal slopes. Therefore, the line perpendicular to 
must have a slope of 
. Knowing that the slope of 
is 
, only 
has a slope of 
.
Lines that are perpendicular have negative reciprocal slopes. Therefore, the line perpendicular to
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Write the equation from the augmented matrix.
Write the equation from the augmented matrix.
Do the first row first and use x and y to represent your variable.
Do the first row first and use x and y to represent your variable.
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Solve for 
in the equation.
Solve for
Solve for x by isolating the variable.
Solve for x by isolating the variable.
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If the equation 
was shifted left three units and up one unit, what is the new equation of the line?
If the equation
If the equation shifts left three units, the 
term will become 
.
The equation shifting up one unit will change the y-intercept of the equation.
Rewrite the equation and distribute to simplify.
The correct equation is: 
If the equation shifts left three units, the
The equation shifting up one unit will change the y-intercept of the equation.
Rewrite the equation and distribute to simplify.
The correct equation is:
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Write the equation of a line that is parallel and two points lower than the line 
.
Write the equation of a line that is parallel and two points lower than the line
Straight-line equations may be written in the slope-intercept form: 
.
In this form, 
equals the slope of the line and 
corresponds to the y-intercept.
The given line has a slope of 
and a y-intercept of positive 
. A line that is parallel to another has the same slope. Therefore, the slope of the new line will have to be 
.
In order to shift a line down, you must change the y-intercept. Since we are moving the line down by 
the y-intercept should be 
because 
.
If we plug those values into the slope-intercept equation, then we have the answer: 
.
Straight-line equations may be written in the slope-intercept form:
In this form,
The given line has a slope of
In order to shift a line down, you must change the y-intercept. Since we are moving the line down by
If we plug those values into the slope-intercept equation, then we have the answer:
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Given the equation 
, which of the following lines are steeper?
Given the equation
Considering that slope (m) is defined as rise over run, you can look that the fractional slopes and determine which are steeper or more flat. For example, 
is equivalent to up one and over 8 while 
is equivalent to up one and over 10. As you can see the slope of the second line "runs" horizontally more than does the first slope and is therefore flatter. Based on this fact one can conclude that the larger the the slope, the steeper the line. So select the largest slope and this is the steepest line. In our case it is 
because 
is steeper (larger) than 
(flatter and a smaller number).
Considering that slope (m) is defined as rise over run, you can look that the fractional slopes and determine which are steeper or more flat. For example,
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The equation 
is shifted eight units downward. Write the new equation.
The equation
Rewrite the equation in slope-intercept format, 
.
Subtract two on both sides.
If the equation shifts eight units down, this means that the y-intercept, 
, would also subtracted eight units.
The correct answer is: 
Rewrite the equation in slope-intercept format,
Subtract two on both sides.
If the equation shifts eight units down, this means that the y-intercept,
The correct answer is:
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Which of the following describes the transformation of the function 
from its parent function 
?
Which of the following describes the transformation of the function
The only differences among the answer choices is the translation. The translation of a function is determined by 
, which represents a horizontal translation h units to the right and k units up. In this case, h = 3 and k = 0, which indicates a translation 3 units to the right.
The only differences among the answer choices is the translation. The translation of a function is determined by
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If the line 
is shifted up two units, and left three units, what is the new equation?
If the line
Vertical shifts will change the y-intercept. Shifting the equation up two units will add two to the y-intercept.
The equation becomes: 
Shifting the equation left three units means that the inner term 
will become 
.
Replace the term.
The equation becomes: 
Simplify this equation by distribution.
The answer is: 
Vertical shifts will change the y-intercept. Shifting the equation up two units will add two to the y-intercept.
The equation becomes:
Shifting the equation left three units means that the inner term
Replace the term.
The equation becomes:
Simplify this equation by distribution.
The answer is:
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Suppose 
is shifted left two units. What is the new equation in slope-intercept form?
Suppose
Rewrite the given standard form equation in slope-intercept format: 
Subtract 
from both sides.
Divide by two on both sides.
Simplify both sides.
If this equation is shifted left two units, the 
will be replaced with 
.
Rewrite the equation and simplify.
The answer is: 
Rewrite the given standard form equation in slope-intercept format:
Subtract
Divide by two on both sides.
Simplify both sides.
If this equation is shifted left two units, the
Rewrite the equation and simplify.
The answer is:
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Shift 
left four units. Write the new equation.
Shift
Simplify the equation given by distributing the integer through the binomial and combine like-terms. This will put the equation in slope intercept form.
Since this equation is shifted left four units, replace 
with 
.
Simplify this equation.
The new equation after the shift is: 
Simplify the equation given by distributing the integer through the binomial and combine like-terms. This will put the equation in slope intercept form.
Since this equation is shifted left four units, replace
Simplify this equation.
The new equation after the shift is:
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If the line 
is shifted up four units, what is the new equation?
If the line
Rewrite the given equation, 
, in standard form to slope intercept form, 
.
Subtract 
from both sides.
Divide by two on both sides.
Simplify the equation.
The vertical shift by four units will shift the y-intercept up four units. Add four to the equation.
The answer is: 
Rewrite the given equation,
Subtract
Divide by two on both sides.
Simplify the equation.
The vertical shift by four units will shift the y-intercept up four units. Add four to the equation.
The answer is:
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Translate the function 
up two units. What is the y-intercept of the new equation?
Translate the function
The equation given is currently in standard form.
Rewrite the equation in slope-intercept form, 
.
Subtract 
on both sides of 
.
Divide by two on both sides.
Simplify the fractions and split the right fraction into two parts.
The equation in slope-intercept form is: 
Apply the translation. If this line is shifted up two units, the y-intercept will be added two.
The answer is: 
The equation given is currently in standard form.
Rewrite the equation in slope-intercept form,
Subtract
Divide by two on both sides.
Simplify the fractions and split the right fraction into two parts.
The equation in slope-intercept form is:
Apply the translation. If this line is shifted up two units, the y-intercept will be added two.
The answer is:
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Transform the equation into slope-intercept form.
Transform the equation into slope-intercept form.
In order to take an action from standard form to slope-intercept form you want to make it of the form:
where 
is the y-intercept (constant/number without a variable attached to it)
is the slope and coefficient of the 
term
and the equation is set equal to 
making it easier to plot graphically.
Given:
I. Isolate 
on one side of the equation. This is done by shifting either 
over to the other side of equation or the 
term and the constant to the other side of the equation. It is generally preferable to shift it to make so the 
term is positive by itself to simply operations and sign mixups. So in this case both 
and 
would be subtracted from both sides of the equation leaving:
II. Now that 
is isolated by itself you want to simplify the equation so there is no coefficient other than 
attached to 
, so in this case it'd mean dividing both sides by 
leaving:
III. Simplify the expression. If you are given fractions that are divisible by each other they can be simplified.
In the case 
is simplified to 
and 
is simplified to 
leaving the final answer of:
In order to take an action from standard form to slope-intercept form you want to make it of the form:
where
and the equation is set equal to
Given:
I. Isolate
II. Now that
III. Simplify the expression. If you are given fractions that are divisible by each other they can be simplified.
In the case
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Shift the line 
right three units. What is the new equation?
Shift the line
Rewrite the given equation 
in standard form to slope-intercept form, 
.
Add 
and subtract three on both sides.
Simplify both sides.
Since the line is shifted three units to the right, the 
term will become 
.
Replace this with the variable in the equation.
Simplify.
The answer is: 
Rewrite the given equation
Add
Simplify both sides.
Since the line is shifted three units to the right, the
Replace this with the variable in the equation.
Simplify.
The answer is:
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Shift the line 
up one unit, and left two units. Write the new equation.
Shift the line
Shifting the line up one unit will result in adding one to the y-intercept.
When the line is shifted left two units, the 
variable must be replaced with the 
term.
Use the distributive property to expand this equation.
The new equation is: 
The answer is: 
Shifting the line up one unit will result in adding one to the y-intercept.
When the line is shifted left two units, the
Use the distributive property to expand this equation.
The new equation is:
The answer is:
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For the function 
we will define a linear transformation 
such that 
. Find the slope and y-intercept of the inverse function 
of 
.
For the function
Use the given function 
to find the linear transformation defined by 
.
First understand that in order to write 
we have to take our function 
and evaluate it for 
and then add 
to the result.
Adding 
,
Now that we have 
, compute the inverse 
. Conventionally we replace the 
notation with the 
notation and solve for 
Therefore,
At this point it's conventional to interchange 
and 
to write the inverse since we want to express it as a function of 
Therefore the slope is 
and the y-intercept is 
Use the given function
First understand that in order to write
Adding
Now that we have
Therefore,
At this point it's conventional to interchange
Therefore the slope is
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Shift the line 
up two units and left three units. What's the new equation?
Shift the line
Shifting the line up two units will add two to the y-intercept.
Shifting the line left three units mean that the x-variable will be replaced with:
Replace 
with the quantity of 
.
Simplify this equation by combining like-terms
The answer is: 
Shifting the line up two units will add two to the y-intercept.
Shifting the line left three units mean that the x-variable will be replaced with:
Replace
Simplify this equation by combining like-terms
The answer is:
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