Examples of direction in the following topics:
-
- When two variables change proportionally, or are directly proportional, to each other, they are said to be in direct variation.
- When two variables change proportionally to each other, they are said to be in direct variation.
- Direct variation is easily illustrated using a linear graph.
- Graph of direct variation with the linear equation y=0.8x.
- The line y=kx is an example of direct variation between variables x and y.
-
- Two variables in direct variation have a linear relationship, while variables in inverse variation do not.
- If $x$ and $y$ are in direct variation,
and $x$ is doubled, then $y$ would also be doubled.
- Direct variation is represented by a linear equation, and can be modeled by graphing a line.
- Inverse variation is
the opposite of direct variation.
- Relate the concept of slope to the concepts of direct and inverse variation
-
- Stretching and shrinking refer to transformations that alter how compact a function looks in the $x$ or $y$ direction.
- This leads to a "stretched" appearance in the vertical direction.
- If the function $f(x)$ is multiplied by a value less than one, all the $y$ values of the equation will decrease, leading to a "shrunken" appearance in the vertical direction.
- This leads to a "shrunken" appearance in the horizontal direction.
- If the independent variable $x$ is multiplied by a value less than one, all the x values of the equation will decrease, leading to a "stretched" appearance in the horizontal direction.
-
- The first coordinate $r$ is the radius or length of the directed line segment from the pole.
- The angle $θ$, measured in radians, indicates the direction of $r$.
- We move counterclockwise from the polar axis by an angle of $θ$,and measure a directed line segment the length of $r$ in the direction of $θ$.
- Adding any number of full turns ($360^{\circ} $ or $2\pi$ radians) to the angular coordinate does not change the corresponding direction.
- Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction.
-
- Before go deeper into the concept of combined variation, it is important to first understand what direct and inverse variation mean.
- Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other.
- If x and y are in direct variation, and x is doubled, then y would also be doubled.
- To have variables that are in combined variation, the equation must have variables that are in both direct and inverse variation, as shown in the example below.
- Apply the techniques learned with direct and inverse variation to combined variation
-
- A translation of a function is a shift in one or more directions.
- A translation moves every point in a function a constant distance in a specified direction.
-
- Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality.
- Now, multiply the same inequality by -3 (remember to change the direction of the symbol):
- This statement also holds true, and it demonstrates how crucial it is to change the direction of the greater than or less than symbol.
- Multiply both sides by -3, remembering to change the direction of the symbol:
-
- Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
- In mathematics, the slope of a line is a number that describes both the direction and the steepness of the line.
- The direction of a line is either increasing, decreasing, horizontal or vertical.
-
- The sign on the coefficient $a$ determines the direction of the parabola.
-
- The arrows on the axes indicate that they extend forever in their respective directions (i.e. infinitely).
- The arrows on the axes indicate that they extend forever in their respective directions (i.e. infinitely).