Linear Functions - Math

A line tangent to a circle at a given point is perpendicular to the radius from the center to that point. That radius, which has endpoints , has slope

.

The line, being perpendicular to this radius, will have slope equal to the opposite of the reciprocal of that of the radius. This slope will be . Since it includes point , we can use the point-slope form of the line to find its equation:

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

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.

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 :

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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 .

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

Solve for x by isolating the variable.

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:

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: .

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).

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:

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.

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:

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:

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:

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: