18 Questions
What does the slope of a line represent?
The rate at which the line raises or falls
If a line has a slope of 0, what type of line is it?
Horizontal line
Which formula is used to calculate the slope between two points on a straight line?
$\frac{y_2 - y_1}{x_2 - x_1}$
If a line has a positive slope, how does it behave from left to right?
It rises
In the slope calculation formula (mathematically), what do (x1, y1) and (x2, y2) represent?
(x1, y1) and (x2, y2) coordinates respectively
What does it mean if a line has an undefined slope?
It is a vertical line
What is the slope-intercept form of an equation where the slope is denoted by 𝑚 and the y-intercept is denoted by 𝑘?
𝑦 = 𝑚𝑥 + 𝑘
In the equation 𝑦 = −5𝑥 + 15, what is the slope of the line?
-5
If a line has a slope of -2 and passes through the point (2, 8), what is the equation of this line in slope-intercept form?
𝑦 = -2𝑥 + 12
If a line has a slope of 0 and a y-intercept of 4, what is the equation of this line?
𝑦 = 0 + 4𝑥
For a non-vertical straight line with slope 𝑚 and containing point (3, 5), what is the slope of the line connecting (3, 5) with any other point (7, 9)?
-2
In the equation 𝑎𝑥 + 𝑏𝑦 = 𝑐, how can you rewrite it to be in slope-intercept form?
$y = \frac{-a}{b}x + \frac{c}{b}$
What does a slope of 3 indicate in a linear function?
The increase in weekly salary associated with each additional unit sold
In a linear function with two variables x and x_2, what does the equation y = f(x, x_2) = ax + ax_2 + a represent?
The dependent variable y as a function of x and x_2
What is the two-point formula for calculating the slope of a line?
Slope = (y2 - y1) / (x2 - x1)
In a linear cost function, why are organizations concerned with costs?
Costs reflect the money flowing out of the organization
How does a linear function y = 300x + 500x_2 + 2500 calculate the weekly salary for a salesperson?
By multiplying the number of units sold by respective commission values
What does it mean when a linear function states that 'a change in the value of y is directly proportional to a change in the variable x'?
Increasing x will always increase y
Study Notes
Equations of Lines
- Equations of the form x = k have no y-intercept, and equations of the form y = k have no x-intercept.
Slope of a Line
- Slope represents the inclination of a line or the rate at which the line rises and falls.
- The slope of a line can be positive, negative, zero, or undefined.
- A positive slope indicates a line that rises from left to right.
- A negative slope indicates a line that falls from left to right.
- A zero slope indicates a horizontal line.
- An undefined slope indicates a vertical line.
- The sign of the slope shows whether the line is falling or rising, and its magnitude shows the steepness of the line.
Two-Point Formula
- The slope of a line can be computed using the two-point formula: Slope = Δy / Δx.
- The formula for the slope of a line connecting two points (x1, y1) and (x2, y2) is: Slope = (y2 - y1) / (x2 - x1).
Slope-Intercept Form
- The slope-intercept form of a linear equation is: y = mx + k, where m is the slope and k is the y-intercept.
- The slope-intercept form can be derived from the general form of a linear equation: ax + by = c.
Determining the Equation of a Straight Line
- If the slope and y-intercept are given, the equation of the line can be written as: y = mx + k.
- If the slope and one point are given, the equation of the line can be written by substituting the known slope and coordinates of the point into the slope-intercept form.
Linear Functions
- A linear function has the form: y = f(x) = ax + a.
- A linear function indicates that a change in the value of y is directly proportional to a change in the value of x.
- Linear functions can have multiple independent variables, such as y = f(x, x2) = ax + ax2 + a.
Linear Cost Function
- Linear cost functions represent the costs associated with an organization, which reflect the money flowing out of the organization.
- Linear cost functions can be used to model the costs of producing and selling products.
Learn about how equations of the form x=k and y=k impact the y-intercept and x-intercept, as well as the concept of slope and its significance in characterizing straight lines. Understand how slope reflects the inclination and steepness of a line, with examples of positive, negative, zero, and undefined slopes.
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