Transformations of Linear Functions - Algebra II

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Question

Which line would never intersect a line with the slope $m=-\frac{3}{2}$ ?

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 $m=-\frac{3}{2}$ . Each answer can be converted to the form or by knowing that in the equation , the slope of the line is simply $-\frac{A}{B}$ . In the correct answer, , the slope would be $-\frac{6}{4}$ , which simplfies to $-\frac{3}{2}$ .

Note the y-intercept is irrelevant to finding the correct answer.

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Question

What is the equation of the line that intersects the point and ?

Answer

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,

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ .

By substituting, we get

$\frac{3-(-3)}{0-(-2)}=\frac{6}{-2}=-3$ 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 :

-->

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Question

Which line is perpendicular to the line ?

Answer

Lines that are perpendicular have negative reciprocal slopes. Therefore, the line perpendicular to must have a slope of . Knowing that the slope of is $-\frac{A}{B}$ , only has a slope of $-\frac{1}{4}$ .

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Question

Write the equation from the augmented matrix.

$[ 4 -6 \left \right | 11 ]$

$\left [ -7 \ 5 \left \right | 13 ]$

Answer

Do the first row first and use x and y to represent your variable.

$\left [ \left 4 \right-6 |\right 11] = 4x - 6y = 11$

$\left [ -7 \5 \left | 13] = -7x + 5y = 13$

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Question

Solve for in the equation.

Answer

Solve for x by isolating the variable.

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Question

If the equation was shifted left three units and up one unit, what is the new equation of the line?

Answer

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:

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Question

Write the equation of a line that is parallel and two points lower than the line .

Answer

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: .

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Question

Given the equation $y=\frac{1}{5}x+8$ , which of the following lines are steeper?

Answer

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, $m=\frac{1}{8}$ is equivalent to up one and over 8 while $m=\frac{1}{10}$ 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 $\frac{1}{4}$ is steeper (larger) than $\frac{1}{5}$ (flatter and a smaller number).

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Question

The equation is shifted eight units downward. Write the new equation.

Answer

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:

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Question

Which of the following describes the transformation of the function $g\left ( x\right )=2\left ( x-3\right )^2$ from its parent function $f\left ( x\right )=x^2$ ?

Answer

The only differences among the answer choices is the translation. The translation of a function is determined by $f\left ( x-h\right )+k$ , 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.

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Question

If the line is shifted up two units, and left three units, what is the new equation?

Answer

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:

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Question

Suppose is shifted left two units. What is the new equation in slope-intercept form?

Answer

Rewrite the given standard form equation in slope-intercept format:

Subtract from both sides.

Divide by two on both sides.

$\frac{2y}{2}=\frac{-2x+3}{2}$

Simplify both sides.

$y=-x+\frac{3}{2}$

If this equation is shifted left two units, the will be replaced with .

Rewrite the equation and simplify.

$y=-(x+2)+\frac{3}{2}$

$y=-x-2+\frac{3}{2} = -x-\frac{4}{2}+\frac{3}{2}=-x-\frac{1}{2}$

The answer is: $y=-x-\frac{1}{2}$

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Question

Shift left four units. Write the new equation.

Answer

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:

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Question

If the line is shifted up four units, what is the new equation?

Answer

Rewrite the given equation, , in standard form to slope intercept form, .

Subtract from both sides.

Divide by two on both sides.

$\frac{2y}{2}=\frac{-x+3}{2}$

Simplify the equation.

$y=-\frac{1}{2}x+\frac{3}{2}$

The vertical shift by four units will shift the y-intercept up four units. Add four to the equation.

$y=-\frac{1}{2}x+\frac{3}{2}+(4)=-\frac{1}{2}x+\frac{3}{2}+\frac{8}{2}$

The answer is: $y=-\frac{1}{2}x+\frac{11}{2}$

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Question

Translate the function up two units. What is the y-intercept of the new equation?

Answer

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.

$\frac{2y}{2}=\frac{-2x+1}{2}$

Simplify the fractions and split the right fraction into two parts.

The equation in slope-intercept form is: $y=-x+\frac{1}{2}$

Apply the translation. If this line is shifted up two units, the y-intercept will be added two.

$\frac{1}{2}+2 = 2\frac{1}{2}$

The answer is: $2\frac{1}{2}$

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Question

Transform the equation into slope-intercept form.

Answer

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:

$y=-\frac{4}{8}x-\frac{2}{8}$

III. Simplify the expression. If you are given fractions that are divisible by each other they can be simplified.

In the case $\frac{4}{8}$ is simplified to $\frac{1}{2}$ and $\frac{2}{8}$ is simplified to $\frac{1}{4}$ leaving the final answer of:

$y=-\frac{1}{2}x-\frac{1}{4}$

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Question

Shift the line right three units. What is the new equation?

Answer

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:

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Question

Shift the line up one unit, and left two units. Write the new equation.

Answer

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:

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Question

For the function we will define a linear transformation such that . Find the slope and y-intercept of the inverse function $g^{-1}(x)$ of .

Answer

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 $g^{-1}(x)$ . Conventionally we replace the notation with the notation and solve for

Therefore,

$x : : : =: : : \frac{y-10}{7}$

$x=\frac{1}{7}y-\frac{10}{7}$

At this point it's conventional to interchange and to write the inverse since we want to express it as a function of

$y =\frac{1}{7}x-\frac{10}{7}$

$g^{-1}(x)=\frac{1}{7}x-\frac{10}{7}$

Therefore the slope is $\frac{1}{7}$ and the y-intercept is $-\frac{10}{7}$

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Question

Shift the line up two units and left three units. What's the new equation?

Answer

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:

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