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Questions and Answers
Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, which of the following formulas correctly calculates the slope, m, of the line?
Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, which of the following formulas correctly calculates the slope, m, of the line?
- $m = \frac{y_1 + y_2}{x_1 + x_2}$
- $m = \frac{x_1 - x_2}{y_2 - y_1}$
- $m = \frac{x_2 - x_1}{y_2 - y_1}$
- $m = \frac{y_2 - y_1}{x_2 - x_1}$ (correct)
A line passes through the points (3, 5) and (5, 9). What is the slope of this line?
A line passes through the points (3, 5) and (5, 9). What is the slope of this line?
- -2
- 2 (correct)
- 0
- Undefined
A line passes through the points (-1, 6) and (-2, 7). What is the slope of this line?
A line passes through the points (-1, 6) and (-2, 7). What is the slope of this line?
- -1 (correct)
- 0
- 1
- Undefined
What is the slope of a horizontal line?
What is the slope of a horizontal line?
What is the slope of a vertical line?
What is the slope of a vertical line?
Two lines are parallel if and only if:
Two lines are parallel if and only if:
Line L1 passes through points (-2, -1) and (1, 3), and line L2 passes through points (4, -10) and (3, -4). Are L1 and L2 parallel?
Line L1 passes through points (-2, -1) and (1, 3), and line L2 passes through points (4, -10) and (3, -4). Are L1 and L2 parallel?
Two lines are perpendicular if and only if:
Two lines are perpendicular if and only if:
Line L1 passes through points (-2, 5) and (4, 2), and line L2 passes through points (-1, -2) and (3, -6). Are lines L1 and L2 perpendicular?
Line L1 passes through points (-2, 5) and (4, 2), and line L2 passes through points (-1, -2) and (3, -6). Are lines L1 and L2 perpendicular?
What is the point-slope form of a line, given a point $(x_1, y_1)$ and a slope m?
What is the point-slope form of a line, given a point $(x_1, y_1)$ and a slope m?
A line passes through the point (1, 3) and has a slope of 2. What is the equation of the line in point-slope form?
A line passes through the point (1, 3) and has a slope of 2. What is the equation of the line in point-slope form?
A line passes through the point (5, 2) and has a slope of $\frac{1}{2}$. What is the equation of the line in point-slope form?
A line passes through the point (5, 2) and has a slope of $\frac{1}{2}$. What is the equation of the line in point-slope form?
A line passes through the points (-3, 2) and (4, -1). What is the equation of the line in point-slope form?
A line passes through the points (-3, 2) and (4, -1). What is the equation of the line in point-slope form?
What is the slope-intercept form of a line?
What is the slope-intercept form of a line?
A line has a slope of 3 and a y-intercept of 7. What is the equation of the line in slope-intercept form?
A line has a slope of 3 and a y-intercept of 7. What is the equation of the line in slope-intercept form?
Given the equation $3x - 4y = 8$, what is the slope, m, and y-intercept, b, of the line?
Given the equation $3x - 4y = 8$, what is the slope, m, and y-intercept, b, of the line?
Which of the following equations represents a line with a y-intercept of 2?
Which of the following equations represents a line with a y-intercept of 2?
Flashcards
Slope of a Line
Slope of a Line
The measure of the steepness of a line. Given two points (x1, y1) and (x2, y2), the slope (m) is calculated as (y2 - y1) / (x2 - x1).
Parallel Lines
Parallel Lines
Lines that have the same slope. They never intersect.
Perpendicular Lines
Perpendicular Lines
Lines that intersect at a right angle (90 degrees). If line 1 has slope m1 and line 2 has slope m2, then the lines are perpendicular if m1 = -1/m2.
Point-Slope Form
Point-Slope Form
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Slope-Intercept Form
Slope-Intercept Form
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Revenue Function
Revenue Function
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Cost Function
Cost Function
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Profit Function
Profit Function
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Break-Even Point
Break-Even Point
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Quadratic Function
Quadratic Function
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Vertex
Vertex
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The pair of supply and demand
The pair of supply and demand
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Study Notes
- These notes cover Cartesian coordinates, slopes, equations of lines, parallel and perpendicular lines, linear functions, quadratic functions, and supply and demand.
The Cartesian Coordinate System and Straight Lines
- Points can be plotted on a Cartesian coordinate system using their x and y coordinates.
Slope of a Line
- Slope (m) is calculated as m = (y2 - y1) / (x2 - x1) given two points (x1, y1) and (x2, y2).
- A positive slope indicates the line moves up from left to right.
- A negative slope indicates the line moves down from left to right.
- A slope of zero indicates a horizontal line.
- An undefined slope indicates a vertical line.
Parallel Lines
- Two distinct lines are parallel if and only if their slopes are equal (m1 = m2).
Equations of Lines
- The point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
- The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
- The equation of a line can be found given a point and a slope by substituting into the point-slope form and rearranging into slope-intercept form.
Slope - Intercept Form
- In the equation y = mx + b, 'm' represents the slope and 'b' represents the y-intercept.
Perpendicular Lines
- Two lines are perpendicular if the product of their slopes is -1 (m1 = -1/m2).
Linear Functions
- Revenue is the money collected from selling items, R(x) = S(x), where S is the selling price per unit and x is the number of items sold.
- Cost function refers to the money spent to produce a certain number of items, C(x) = Cx + F, Fixed price is F and C is cost per unit and x represents the number of items produced.
- Profit is the revenue minus the cost, P(x) = R(x) - C(x).
Quadratic Functions
- A quadratic function is in the form f(x) = ax^2 + bx + c (a ≠0).
- The vertex of a parabola is given by (-b/2a, f(-b/2a)).
- The axis of symmetry is x = -b/2a.
- To find x-intercepts, set y = 0 and solve for x.
- Y intercepts are calculated by setting x=0
Break-Even Point
- Break-even occurs when Profit = 0 or R(x) = C(x).
Demand and Supply Curves
- Supply function has an upward curve
- Demand function has a downward curve
- Equilibrium occurs where demand equals supply function
- Find equilibrium price by substituting 'x' value in the demand or supply equation
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